Results of Elliptic Integrals I

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.2.E2	${\displaystyle{\displaystyle r(s,t)=\frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s}}}$ r(s,t) = \frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s}		r(s , t) = ((p[1]+ p[2]*s)*(p[3]- p[4]*s)*s)/((p[3]+ p[4]*s)*(p[3]- p[4]*s)*s)	r[s , t] == Divide[(Subscript[p, 1]+ Subscript[p, 2]*s)*(Subscript[p, 3]- Subscript[p, 4]*s)*s,(Subscript[p, 3]+ Subscript[p, 4]*s)*(Subscript[p, 3]- Subscript[p, 4]*s)*s]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.2.E4	${\displaystyle{\displaystyle F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}}}$ \incellintFk@{\phi}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}}		EllipticF(sin(phi), k) = int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)	EllipticF[\[Phi], (k)^2] == Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None]	Failure	Aborted	Failed [6 / 30] Result: Float(infinity) Test Values: {phi = -2, k = 1} Result: .2e-9-.5175477340*I Test Values: {phi = -2, k = 2} ... skip entries to safe data	Skipped - Because timed out
19.2.E4	${\displaystyle{\displaystyle\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}}}$ \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}		int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi) = int((1)/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))	Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[1,Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]	Failure	Aborted	Failed [6 / 30] Result: Float(-infinity) Test Values: {phi = -2, k = 1} Result: -.2e-9+.5175477340*I Test Values: {phi = -2, k = 2} ... skip entries to safe data	Skipped - Because timed out
19.2.E5	${\displaystyle{\displaystyle E\left(\phi,k\right)=\int_{0}^{\phi}\sqrt{1-k^{2}{\sin^{2}}\theta}\mathrm{d}\theta\\ }}$ \incellintEk@{\phi}{k} = \int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\		EllipticE(sin(phi), k) = int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi)	EllipticE[\[Phi], (k)^2] == Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.2.E5	${\displaystyle{\displaystyle\int_{0}^{\phi}\sqrt{1-k^{2}{\sin^{2}}\theta}\mathrm{d}\theta\\ =\int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\mathrm{d}t}}$ \int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}@@{\theta}}\diff{\theta}\\ = \int_{0}^{\sin@@{\phi}}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\diff{t}		int(sqrt(1 - (k)^(2)* (sin(theta))^(2)), theta = 0..phi) = int((sqrt(1 - (k)^(2)* (t)^(2)))/(sqrt(1 - (t)^(2))), t = 0..sin(phi))	Integrate[Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[Sqrt[1 - (k)^(2)* (t)^(2)],Sqrt[1 - (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.2.E6	${\displaystyle{\displaystyle D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin^{2}}\theta\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}}}$ \incellintDk@{\phi}{k} = \int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}}		(EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2 = int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi)	Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4] == Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.2.E6	${\displaystyle{\displaystyle\int_{0}^{\phi}\frac{{\sin^{2}}\theta\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}}}$ \int_{0}^{\phi}\frac{\sin^{2}@@{\theta}\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}} = \int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}		int(((sin(theta))^(2))/(sqrt(1 - (k)^(2)* (sin(theta))^(2))), theta = 0..phi) = int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi))	Integrate[Divide[(Sin[\[Theta]])^(2),Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.2.E6	${\displaystyle{\displaystyle\int_{0}^{\sin\phi}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))/k^{2}}}$ \int_{0}^{\sin@@{\phi}}\frac{t^{2}\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}} = (\incellintFk@{\phi}{k}-\incellintEk@{\phi}{k})/k^{2}		int(((t)^(2))/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))), t = 0..sin(phi)) = (EllipticF(sin(phi), k)- EllipticE(sin(phi), k))/(k)^(2)	Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]], {t, 0, Sin[\[Phi]]}, GenerateConditions->None] == (EllipticF[\[Phi], (k)^2]- EllipticE[\[Phi], (k)^2])/(k)^(2)	Failure	Aborted	Successful [Tested: 0]	Skipped - Because timed out
19.2.E7	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}(1-\alpha^{2}{\sin^{2}}\theta)}}}$ \incellintPik@{\phi}{\alpha^{2}}{k} = \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})}		EllipticPi(sin(phi), (alpha)^(2), k) = int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))*(1 - (alpha)^(2)* (sin(theta))^(2))), theta = 0..phi)	EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]*(1 - \[Alpha]^(2)* (Sin[\[Theta]])^(2))], {\[Theta], 0, \[Phi]}, GenerateConditions->None]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.2.E7	${\displaystyle{\displaystyle\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}(1-\alpha^{2}{\sin^{2}}\theta)}=\int_{0}^{\sin\phi}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}}}$ \int_{0}^{\phi}\frac{\diff{\theta}}{\sqrt{1-k^{2}\sin^{2}@@{\theta}}(1-\alpha^{2}\sin^{2}@@{\theta})} = \int_{0}^{\sin@@{\phi}}\frac{\diff{t}}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}		int((1)/(sqrt(1 - (k)^(2)* (sin(theta))^(2))*(1 - (alpha)^(2)* (sin(theta))^(2))), theta = 0..phi) = int((1)/(sqrt(1 - (t)^(2))*sqrt(1 - (k)^(2)* (t)^(2))*(1 - (alpha)^(2)* (t)^(2))), t = 0..sin(phi))	Integrate[Divide[1,Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]*(1 - \[Alpha]^(2)* (Sin[\[Theta]])^(2))], {\[Theta], 0, \[Phi]}, GenerateConditions->None] == Integrate[Divide[1,Sqrt[1 - (t)^(2)]*Sqrt[1 - (k)^(2)* (t)^(2)]*(1 - \[Alpha]^(2)* (t)^(2))], {t, 0, Sin[\[Phi]]}, GenerateConditions->None]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.2#Ex1	${\displaystyle{\displaystyle K\left(k\right)=F\left(\pi/2,k\right)}}$ \compellintKk@{k} = \incellintFk@{\pi/2}{k}		EllipticK(k) = EllipticF(sin(Pi/2), k)	EllipticK[(k)^2] == EllipticF[Pi/2, (k)^2]	Successful	Successful	-	Successful [Tested: 3]
19.2#Ex2	${\displaystyle{\displaystyle E\left(k\right)=E\left(\pi/2,k\right)}}$ \compellintEk@{k} = \incellintEk@{\pi/2}{k}		EllipticE(k) = EllipticE(sin(Pi/2), k)	EllipticE[(k)^2] == EllipticE[Pi/2, (k)^2]	Successful	Successful	-	Successful [Tested: 3]
19.2#Ex3	${\displaystyle{\displaystyle D\left(k\right)=D\left(\pi/2,k\right)}}$ \compellintDk@{k} = \incellintDk@{\pi/2}{k}		(EllipticK(k) - EllipticE(k))/(k)^2 = (EllipticF(sin(Pi/2), k) - EllipticE(sin(Pi/2), k))/(k)^2	Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[EllipticF[Pi/2, (k)^2] - EllipticE[Pi/2, (k)^2], (k)^4]	Successful	Successful	-	Successful [Tested: 3]
19.2#Ex3	${\displaystyle{\displaystyle D\left(\pi/2,k\right)=(K\left(k\right)-E\left(k\right))/k^{2}}}$ \incellintDk@{\pi/2}{k} = (\compellintKk@{k}-\compellintEk@{k})/k^{2}		(EllipticF(sin(Pi/2), k) - EllipticE(sin(Pi/2), k))/(k)^2 = (EllipticK(k)- EllipticE(k))/(k)^(2)	Divide[EllipticF[Pi/2, (k)^2] - EllipticE[Pi/2, (k)^2], (k)^4] == (EllipticK[(k)^2]- EllipticE[(k)^2])/(k)^(2)	Successful	Failure	-	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-0.08185805455243832, 0.4541460103381725] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.2#Ex4	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\Pi\left(\pi/2,\alpha^{2},k\right)}}$ \compellintPik@{\alpha^{2}}{k} = \incellintPik@{\pi/2}{\alpha^{2}}{k}		EllipticPi((alpha)^(2), k) = EllipticPi(sin(Pi/2), (alpha)^(2), k)	EllipticPi[\[Alpha]^(2), (k)^2] == EllipticPi[\[Alpha]^(2), Pi/2,(k)^2]	Successful	Successful	-	Successful [Tested: 9]
19.2#Ex5	${\displaystyle{\displaystyle{K^{\prime}}\left(k\right)=K\left(k^{\prime}\right)}}$ \ccompellintKk@{k} = \compellintKk@{k^{\prime}}		EllipticCK(k) = EllipticK(sqrt(1 - (k)^(2)))	EllipticK[1-(k)^2] == EllipticK[(Sqrt[1 - (k)^(2)])^2]	Successful	Successful	-	Successful [Tested: 3]
19.2#Ex6	${\displaystyle{\displaystyle{E^{\prime}}\left(k\right)=E\left(k^{\prime}\right)}}$ \ccompellintEk@{k} = \compellintEk@{k^{\prime}}		EllipticCE(k) = EllipticE(sqrt(1 - (k)^(2)))	EllipticE[1-(k)^2] == EllipticE[(Sqrt[1 - (k)^(2)])^2]	Successful	Successful	-	Successful [Tested: 3]
19.2#Ex7	${\displaystyle{\displaystyle k^{\prime}=\sqrt{1-k^{2}}}}$ k^{\prime} = \sqrt{1-k^{2}}		sqrt(1 - (k)^(2)) = sqrt(1 - (k)^(2))	Sqrt[1 - (k)^(2)] == Sqrt[1 - (k)^(2)]	Successful	Successful	-	Successful [Tested: 3]
19.2#Ex8	${\displaystyle{\displaystyle F\left(m\pi+\phi,k\right)=2mK\left(k\right)+F\left(\phi,k\right)}}$ \incellintFk@{m\pi+\phi}{k} = 2m\compellintKk@{k}+\incellintFk@{\phi}{k}		EllipticF(sin(m*Pi + phi), k) = 2*m*EllipticK(k)+ EllipticF(sin(phi), k)	EllipticF[m*Pi + \[Phi], (k)^2] == 2*m*EllipticK[(k)^2]+ EllipticF[\[Phi], (k)^2]	Failure	Failure	Error	Failed [30 / 90] Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.2#Ex8	${\displaystyle{\displaystyle F\left(m\pi-\phi,k\right)=2mK\left(k\right)-F\left(\phi,k\right)}}$ \incellintFk@{m\pi-\phi}{k} = 2m\compellintKk@{k}-\incellintFk@{\phi}{k}		EllipticF(sin(m*Pi - phi), k) = 2*m*EllipticK(k)- EllipticF(sin(phi), k)	EllipticF[m*Pi - \[Phi], (k)^2] == 2*m*EllipticK[(k)^2]- EllipticF[\[Phi], (k)^2]	Failure	Failure	Error	Failed [30 / 90] Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.2#Ex9	${\displaystyle{\displaystyle E\left(m\pi+\phi,k\right)=2mE\left(k\right)+E\left(\phi,k\right)}}$ \incellintEk@{m\pi+\phi}{k} = 2m\compellintEk@{k}+\incellintEk@{\phi}{k}		EllipticE(sin(m*Pi + phi), k) = 2*m*EllipticE(k)+ EllipticE(sin(phi), k)	EllipticE[m*Pi + \[Phi], (k)^2] == 2*m*EllipticE[(k)^2]+ EllipticE[\[Phi], (k)^2]	Failure	Failure	Failed [90 / 90] Result: -3.717960670-.6751929261*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 1} Result: -4.000000000-.3e-9*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 2} ... skip entries to safe data	Successful [Tested: 90]
19.2#Ex9	${\displaystyle{\displaystyle E\left(m\pi-\phi,k\right)=2mE\left(k\right)-E\left(\phi,k\right)}}$ \incellintEk@{m\pi-\phi}{k} = 2m\compellintEk@{k}-\incellintEk@{\phi}{k}		EllipticE(sin(m*Pi - phi), k) = 2*m*EllipticE(k)- EllipticE(sin(phi), k)	EllipticE[m*Pi - \[Phi], (k)^2] == 2*m*EllipticE[(k)^2]- EllipticE[\[Phi], (k)^2]	Failure	Failure	Failed [90 / 90] Result: -.2820393315+.6751929264*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 1} Result: -4.000000000-.4e-9*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1, m = 2} ... skip entries to safe data	Successful [Tested: 90]
19.2#Ex10	${\displaystyle{\displaystyle D\left(m\pi+\phi,k\right)=2mD\left(k\right)+D\left(\phi,k\right)}}$ \incellintDk@{m\pi+\phi}{k} = 2m\compellintDk@{k}+\incellintDk@{\phi}{k}		(EllipticF(sin(m*Pi + phi), k) - EllipticE(sin(m*Pi + phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 + (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2	Divide[EllipticF[m*Pi + \[Phi], (k)^2] - EllipticE[m*Pi + \[Phi], (k)^2], (k)^4] == 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]+ Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]	Failure	Failure	Error	Failed [30 / 90] Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.2#Ex10	${\displaystyle{\displaystyle D\left(m\pi-\phi,k\right)=2mD\left(k\right)-D\left(\phi,k\right)}}$ \incellintDk@{m\pi-\phi}{k} = 2m\compellintDk@{k}-\incellintDk@{\phi}{k}		(EllipticF(sin(m*Pi - phi), k) - EllipticE(sin(m*Pi - phi), k))/(k)^2 = 2*m*(EllipticK(k) - EllipticE(k))/(k)^2 - (EllipticF(sin(phi), k) - EllipticE(sin(phi), k))/(k)^2	Divide[EllipticF[m*Pi - \[Phi], (k)^2] - EllipticE[m*Pi - \[Phi], (k)^2], (k)^4] == 2*m*Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4]- Divide[EllipticF[\[Phi], (k)^2] - EllipticE[\[Phi], (k)^2], (k)^4]	Failure	Failure	Error	Failed [30 / 90] Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[k, 1], Rule[m, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.2.E16	${\displaystyle{\displaystyle\int_{0}^{\operatorname{arctan}x}\frac{\mathrm{d}\theta}{({\cos^{2}}\theta+p{\sin^{2}}\theta)\sqrt{{\cos^{2}}\theta+k_{c}^{2}{\sin^{2}}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right)}}$ \int_{0}^{\atan@@{x}}\frac{\diff{\theta}}{(\cos^{2}@@{\theta}+p\sin^{2}@@{\theta})\sqrt{\cos^{2}@@{\theta}+k_{c}^{2}\sin^{2}@@{\theta}}} = \incellintPik@{\atan@@{x}}{1-p}{k}	${\displaystyle{\displaystyle x^{2}\neq-1/p}}$	int((1)/(((cos(theta))^(2)+ p*(sin(theta))^(2))*sqrt((cos(theta))^(2)+ (k[c])^(2)*(sin(theta))^(2))), theta = 0..arctan(x)) = EllipticPi(sin(arctan(x)), 1 - p, k)	Integrate[Divide[1,((Cos[\[Theta]])^(2)+ p*(Sin[\[Theta]])^(2))*Sqrt[(Cos[\[Theta]])^(2)+ (Subscript[k, c])^(2)*(Sin[\[Theta]])^(2)]], {\[Theta], 0, ArcTan[x]}, GenerateConditions->None] == EllipticPi[1 - p, ArcTan[x],(k)^2]	Error	Aborted	-	Skipped - Because timed out
19.2.E17	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)}}}$ \CarlsonellintRC@{x}{y} = \frac{1}{2}\int_{0}^{\infty}\frac{\diff{t}}{\sqrt{t+x}(t+y)}		Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,2]*Integrate[Divide[1,Sqrt[t + x]*(t + y)], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [12 / 18] Result: Complex[-1.0177225554447185, 0.0] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.2.E18	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{1}{\sqrt{y-x}}\operatorname{arctan}\sqrt{\frac{y-x}{x}}}}$ \CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}}	${\displaystyle{\displaystyle 0\leq x,x<y}}$	Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,Sqrt[y - x]]*ArcTan[Sqrt[Divide[y - x,x]]]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Successful [Tested: 3]
19.2.E18	${\displaystyle{\displaystyle\frac{1}{\sqrt{y-x}}\operatorname{arctan}\sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\operatorname{arccos}\sqrt{x/y}}}$ \frac{1}{\sqrt{y-x}}\atan@@{\sqrt{\frac{y-x}{x}}} = \frac{1}{\sqrt{y-x}}\acos@@{\sqrt{x/y}}	${\displaystyle{\displaystyle 0\leq x,x<y}}$	(1)/(sqrt(y - x))*arctan(sqrt((y - x)/(x))) = (1)/(sqrt(y - x))*arccos(sqrt(x/y))	Divide[1,Sqrt[y - x]]*ArcTan[Sqrt[Divide[y - x,x]]] == Divide[1,Sqrt[y - x]]*ArcCos[Sqrt[x/y]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
19.2.E19	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x-y}{x}}}}$ \CarlsonellintRC@{x}{y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}}	${\displaystyle{\displaystyle 0<y,y<x}}$	Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x - y,x]]]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Successful [Tested: 3]
19.2.E19	${\displaystyle{\displaystyle\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}}}}$ \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x-y}{x}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}}}	${\displaystyle{\displaystyle 0<y,y<x}}$	(1)/(sqrt(x - y))*arctanh(sqrt((x - y)/(x))) = (1)/(sqrt(x - y))*ln((sqrt(x)+sqrt(x - y))/(sqrt(y)))	Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x - y,x]]] == Divide[1,Sqrt[x - y]]*Log[Divide[Sqrt[x]+Sqrt[x - y],Sqrt[y]]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
19.2.E20	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=\sqrt{\frac{x}{x-y}}R_{C}\left(x-y,-y\right)}}$ \CarlsonellintRC@{x}{y} = \sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y}	${\displaystyle{\displaystyle y<0,0\leq x}}$	Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Sqrt[Divide[x,x - y]]*1/Sqrt[- y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x - y)/(- y)]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Failed [9 / 9] Result: Complex[-1.0177225554447187, -0.906899682117109] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-1.862459718905424, -1.1107207345395915] Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.2.E20	${\displaystyle{\displaystyle\sqrt{\frac{x}{x-y}}R_{C}\left(x-y,-y\right)=\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x}{x-y}}}}$ \sqrt{\frac{x}{x-y}}\CarlsonellintRC@{x-y}{-y} = \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}}	${\displaystyle{\displaystyle y<0,0\leq x}}$	Error	Sqrt[Divide[x,x - y]]*1/Sqrt[- y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x - y)/(- y)] == Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x,x - y]]]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Successful [Tested: 9]
19.2.E20	${\displaystyle{\displaystyle\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}}}}$ \frac{1}{\sqrt{x-y}}\atanh@@{\sqrt{\frac{x}{x-y}}} = \frac{1}{\sqrt{x-y}}\ln@@{\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}}}	${\displaystyle{\displaystyle y<0,0\leq x}}$	(1)/(sqrt(x - y))*arctanh(sqrt((x)/(x - y))) = (1)/(sqrt(x - y))*ln((sqrt(x)+sqrt(x - y))/(sqrt(- y)))	Divide[1,Sqrt[x - y]]*ArcTanh[Sqrt[Divide[x,x - y]]] == Divide[1,Sqrt[x - y]]*Log[Divide[Sqrt[x]+Sqrt[x - y],Sqrt[- y]]]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
19.2.E21	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=\int_{0}^{1}(v^{2}x+(1-v^{2})y)^{-1/2}\mathrm{d}v}}$ \CarlsonellintRC@{x}{y} = \int_{0}^{1}(v^{2}x+(1-v^{2})y)^{-1/2}\diff{v}		Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Integrate[((v)^(2)* x +(1 - (v)^(2))*y)^(- 1/2), {v, 0, 1}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.2.E22	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,x{\cos^{2}}\theta+y{\sin^{2}}\theta\right)\mathrm{d}\theta}}$ \CarlsonellintRC@{x}{y} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{x\cos^{2}@@{\theta}+y\sin^{2}@@{\theta}}\diff{\theta}		Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == Divide[2,Pi]*Integrate[1/Sqrt[x*(Cos[\[Theta]])^(2)+ y*(Sin[\[Theta]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(y)/(x*(Cos[\[Theta]])^(2)+ y*(Sin[\[Theta]])^(2))], {\[Theta], 0, Pi/2}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.4#Ex1	${\displaystyle{\displaystyle\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)}{k{k^{\prime}}^{2}}}}$ \deriv{\compellintKk@{k}}{k} = \frac{\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k}}{k{k^{\prime}}^{2}}		diff(EllipticK(k), k) = (EllipticE(k)-1 - (k)^(2)*EllipticK(k))/(k*1 - (k)^(2))	D[EllipticK[(k)^2], k] == Divide[EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2],k*1 - (k)^(2)]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.4717549813624253, 3.1435959698369205] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.4#Ex2	${\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}{\mathrm{d}k}=kK\left(k\right)}}$ \deriv{(\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k})}{k} = k\compellintKk@{k}		diff(EllipticE(k)-1 - (k)^(2)*EllipticK(k), k) = k*EllipticK(k)	D[EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2], k] == k*EllipticK[(k)^2]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-3.3189229307917216, 6.419990143492479] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.4#Ex3	${\displaystyle{\displaystyle\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-K\left(k\right)}{k}}}$ \deriv{\compellintEk@{k}}{k} = \frac{\compellintEk@{k}-\compellintKk@{k}}{k}		diff(EllipticE(k), k) = (EllipticE(k)- EllipticK(k))/(k)	D[EllipticE[(k)^2], k] == Divide[EllipticE[(k)^2]- EllipticK[(k)^2],k]	Successful	Successful	-	Successful [Tested: 3]
19.4#Ex4	${\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}=-\frac{kE\left(k\right)}{{k^{\prime}}^{2}}}}$ \deriv{(\compellintEk@{k}-\compellintKk@{k})}{k} = -\frac{k\compellintEk@{k}}{{k^{\prime}}^{2}}		diff(EllipticE(k)- EllipticK(k), k) = -(k*EllipticE(k))/(1 - (k)^(2))	D[EllipticE[(k)^2]- EllipticK[(k)^2], k] == -Divide[k*EllipticE[(k)^2],1 - (k)^(2)]	Successful	Successful	-	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.4.E3	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}}}$ \deriv[2]{\compellintEk@{k}}{k} = -\frac{1}{k}\deriv{\compellintKk@{k}}{k}		diff(EllipticE(k), [k$(2)]) = -(1)/(k)*diff(EllipticK(k), k)	D[EllipticE[(k)^2], {k, 2}] == -Divide[1,k]*D[EllipticK[(k)^2], k]	Successful	Successful	-	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.4.E3	${\displaystyle{\displaystyle-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-E\left(k\right)}{k^{2}{k^{\prime}}^{2}}}}$ -\frac{1}{k}\deriv{\compellintKk@{k}}{k} = \frac{{k^{\prime}}^{2}\compellintKk@{k}-\compellintEk@{k}}{k^{2}{k^{\prime}}^{2}}		-(1)/(k)*diff(EllipticK(k), k) = (1 - (k)^(2)*EllipticK(k)- EllipticE(k))/((k)^(2)*1 - (k)^(2))	-Divide[1,k]*D[EllipticK[(k)^2], k] == Divide[1 - (k)^(2)*EllipticK[(k)^2]- EllipticE[(k)^2],(k)^(2)*1 - (k)^(2)]	Error	Failure	-	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: DirectedInfinity[] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.4.E4	${\displaystyle{\displaystyle\frac{\partial\Pi\left(\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{\prime}}^{2}\Pi\left(\alpha^{2},k\right))}}$ \pderiv{\compellintPik@{\alpha^{2}}{k}}{k} = \frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(\compellintEk@{k}-{k^{\prime}}^{2}\compellintPik@{\alpha^{2}}{k})		diff(EllipticPi((alpha)^(2), k), k) = (k)/(1 - (k)^(2)*((k)^(2)- (alpha)^(2)))*(EllipticE(k)-1 - (k)^(2)*EllipticPi((alpha)^(2), k))	D[EllipticPi[\[Alpha]^(2), (k)^2], k] == Divide[k,1 - (k)^(2)*((k)^(2)- \[Alpha]^(2))]*(EllipticE[(k)^2]-1 - (k)^(2)*EllipticPi[\[Alpha]^(2), (k)^2])	Failure	Failure	Error	Failed [9 / 9] Result: Indeterminate Test Values: {Rule[k, 1], Rule[α, 1.5]} Result: Complex[0.38994760629924174, 1.2322724929931343] Test Values: {Rule[k, 2], Rule[α, 1.5]} ... skip entries to safe data
19.4.E5	${\displaystyle{\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}={\frac{E\left(\phi,k\right)-{k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}^{2}}-\frac{k\sin\phi\cos\phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin^{2}}\phi}}}}}$ \pderiv{\incellintFk@{\phi}{k}}{k} = {\frac{\incellintEk@{\phi}{k}-{k^{\prime}}^{2}\incellintFk@{\phi}{k}}{k{k^{\prime}}^{2}}-\frac{k\sin@@{\phi}\cos@@{\phi}}{{k^{\prime}}^{2}\sqrt{1-k^{2}\sin^{2}@@{\phi}}}}		diff(EllipticF(sin(phi), k), k) = (EllipticE(sin(phi), k)-1 - (k)^(2)*EllipticF(sin(phi), k))/(k*1 - (k)^(2))-(k*sin(phi)*cos(phi))/(1 - (k)^(2)*sqrt(1 - (k)^(2)* (sin(phi))^(2)))	D[EllipticF[\[Phi], (k)^2], k] == Divide[EllipticE[\[Phi], (k)^2]-1 - (k)^(2)*EllipticF[\[Phi], (k)^2],k*1 - (k)^(2)]-Divide[k*Sin[\[Phi]]*Cos[\[Phi]],1 - (k)^(2)*Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]]	Failure	Failure	Failed [30 / 30] Result: Float(infinity)+Float(infinity)*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -1.296981010-1.781988683*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [30 / 30] Result: Indeterminate Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.2927667883728842, -0.7915995039632082] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.4.E6	${\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F\left(\phi,k\right)}{k}}}$ \pderiv{\incellintEk@{\phi}{k}}{k} = \frac{\incellintEk@{\phi}{k}-\incellintFk@{\phi}{k}}{k}		diff(EllipticE(sin(phi), k), k) = (EllipticE(sin(phi), k)- EllipticF(sin(phi), k))/(k)	D[EllipticE[\[Phi], (k)^2], k] == Divide[EllipticE[\[Phi], (k)^2]- EllipticF[\[Phi], (k)^2],k]	Successful	Successful	-	Successful [Tested: 30]
19.4.E7	${\displaystyle{\displaystyle\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k\right)-{k^{\prime}}^{2}\Pi\left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}\right)}}$ \pderiv{\incellintPik@{\phi}{\alpha^{2}}{k}}{k} = \frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({\incellintEk@{\phi}{k}-{k^{\prime}}^{2}\incellintPik@{\phi}{\alpha^{2}}{k}}-\frac{k^{2}\sin@@{\phi}\cos@@{\phi}}{\sqrt{1-k^{2}\sin^{2}@@{\phi}}}\right)		diff(EllipticPi(sin(phi), (alpha)^(2), k), k) = (k)/(1 - (k)^(2)*((k)^(2)- (alpha)^(2)))*(EllipticE(sin(phi), k)-1 - (k)^(2)*EllipticPi(sin(phi), (alpha)^(2), k)-((k)^(2)* sin(phi)*cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2))))	D[EllipticPi[\[Alpha]^(2), \[Phi],(k)^2], k] == Divide[k,1 - (k)^(2)*((k)^(2)- \[Alpha]^(2))]*(EllipticE[\[Phi], (k)^2]-1 - (k)^(2)*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]-Divide[(k)^(2)* Sin[\[Phi]]*Cos[\[Phi]],Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]])	Failure	Aborted	Failed [90 / 90] Result: Float(undefined)+Float(undefined)*I Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -5.135398794+1.052011331*I Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [90 / 90] Result: Indeterminate Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.1264235284707635, -0.9763567309038728] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.4.E8	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k\sin\phi\cos\phi}{(1-k^{2}{\sin^{2}}\phi)^{3/2}}}}$ (k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)\incellintFk@{\phi}{k} = \frac{-k\sin@@{\phi}\cos@@{\phi}}{(1-k^{2}\sin^{2}@@{\phi})^{3/2}}		(k*1 - (k)^(2)*(D[k])^(2)+(1 - 3*(k)^(2))*D[k]- k)*EllipticF(sin(phi), k) = (- k*sin(phi)*cos(phi))/((1 - (k)^(2)* (sin(phi))^(2))^(3/2))	(k*1 - (k)^(2)*(Subscript[D, k])^(2)+(1 - 3*(k)^(2))*Subscript[D, k]- k)*EllipticF[\[Phi], (k)^2] == Divide[- k*Sin[\[Phi]]*Cos[\[Phi]],(1 - (k)^(2)* (Sin[\[Phi]])^(2))^(3/2)]	Error	Failure	-	Failed [300 / 300] Result: Plus[Complex[0.4174282354972822, 0.36074991075375373], Times[Complex[0.43180375739814203, 0.27142936483528934], Plus[Complex[-0.8660254037844387, -0.49999999999999994], Times[Complex[-0.12500000000000003, -0.21650635094610965], D]]]] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Complex[-0.38000132033999284, 0.977947559972491], Times[Complex[0.3965687056216178, 0.33175091278780894], Plus[Complex[-4.763139720814413, -2.7499999999999996], Times[Complex[-0.5000000000000001, -0.8660254037844386], D]]]] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.4.E9	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac{k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}}}$ (k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)\incellintEk@{\phi}{k} = \frac{k\sin@@{\phi}\cos@@{\phi}}{\sqrt{1-k^{2}\sin^{2}@@{\phi}}}		(k*1 - (k)^(2)*(D[k])^(2)+1 - (k)^(2)*D[k]+ k)*EllipticE(sin(phi), k) = (k*sin(phi)*cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2)))	(k*1 - (k)^(2)*(Subscript[D, k])^(2)+1 - (k)^(2)*Subscript[D, k]+ k)*EllipticE[\[Phi], (k)^2] == Divide[k*Sin[\[Phi]]*Cos[\[Phi]],Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]]	Error	Failure	-	Failed [300 / 300] Result: Plus[Complex[-0.4327885168580316, -0.2292976446734403], Times[Complex[0.43278851685803155, 0.22929764467344024], Plus[Complex[2.566987298107781, -0.24999999999999997], Times[Complex[-0.12500000000000003, -0.21650635094610965], D]]]] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Complex[-0.6011783848834926, -0.7526006723022071], Times[Complex[0.44208095936294645, 0.16535187593702125], Plus[Complex[3.2679491924311224, -0.9999999999999999], Times[Complex[-0.5000000000000001, -0.8660254037844386], D]]]] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.5.E1	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}}}$ \compellintKk@{k} = \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m}		EllipticK(k) = (Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)	EllipticK[(k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 3]
19.5.E1	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}$ \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m} = \frac{\pi}{2}\genhyperF{2}{1}@{\tfrac{1}{2},\tfrac{1}{2}}{1}{k^{2}}		(Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(2)*hypergeom([(1)/(2),(1)/(2)], [1], (k)^(2))	Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*HypergeometricPFQ[{Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]	Failure	Successful	Failed [3 / 3] Result: Float(infinity)+Float(infinity)*I Test Values: {k = 1} Result: Float(infinity)+1.078257824*I Test Values: {k = 2} ... skip entries to safe data	Successful [Tested: 3]
19.5.E2	${\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}}}$ \compellintEk@{k} = \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{-\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m}		EllipticE(k) = (Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)	EllipticE[(k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [2 / 3] Result: Float(infinity)+1.343854231*I Test Values: {k = 2} Result: Float(infinity)+2.498348128*I Test Values: {k = 3}	Successful [Tested: 3]
19.5.E2	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}$ \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{-\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m} = \frac{\pi}{2}\genhyperF{2}{1}@{-\tfrac{1}{2},\tfrac{1}{2}}{1}{k^{2}}		(Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(2)*hypergeom([-(1)/(2),(1)/(2)], [1], (k)^(2))	Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*HypergeometricPFQ[{-Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]	Failure	Successful	Failed [2 / 3] Result: Float(-infinity)-1.343854232*I Test Values: {k = 2} Result: Float(-infinity)-2.498348127*I Test Values: {k = 3}	Successful [Tested: 3]
19.5.E3	${\displaystyle{\displaystyle D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}}}$ \compellintDk@{k} = \frac{\pi}{4}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{3}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{(m+1)!\;m!}k^{2m}		(EllipticK(k) - EllipticE(k))/(k)^2 = (Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity)	Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-0.08185805455243848, 0.4541460103381727] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.5.E3	${\displaystyle{\displaystyle\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{}_{2}F_{1}}\left(\tfrac{3}{2},\tfrac{1}{2};2;k^{2}\right)}}$ \frac{\pi}{4}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{3}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{(m+1)!\;m!}k^{2m} = \frac{\pi}{4}\genhyperF{2}{1}@{\tfrac{3}{2},\tfrac{1}{2}}{2}{k^{2}}		(Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(4)*hypergeom([(3)/(2),(1)/(2)], [2], (k)^(2))	Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,4]*HypergeometricPFQ[{Divide[3,2],Divide[1,2]}, {2}, (k)^(2)]	Failure	Successful	Failed [3 / 3] Result: Float(infinity)+Float(infinity)*I Test Values: {k = 1} Result: Float(infinity)+.6055280139*I Test Values: {k = 2} ... skip entries to safe data	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.5.E4	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{\left(\tfrac{1}{2}\right)_{m}}}{m!}k^{2m}\alpha^{2n-2m}}}$ \compellintPik@{\alpha^{2}}{k} = \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{n}}{n!}\sum_{m=0}^{n}\frac{\Pochhammersym{\tfrac{1}{2}}{m}}{m!}k^{2m}\alpha^{2n-2m}		EllipticPi((alpha)^(2), k) = (Pi)/(2)*sum((pochhammer((1)/(2), n))/(factorial(n))*sum((pochhammer((1)/(2), m))/(factorial(m))*(k)^(2*m)* (alpha)^(2*n - 2*m), m = 0..n), n = 0..infinity)	EllipticPi[\[Alpha]^(2), (k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], n],(n)!]*Sum[Divide[Pochhammer[Divide[1,2], m],(m)!]*(k)^(2*m)* \[Alpha]^(2*n - 2*m), {m, 0, n}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Error	Skipped - Because timed out
19.5.E4	${\displaystyle{\displaystyle\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{\left(\tfrac{1}{2}\right)_{m}}}{m!}k^{2m}\alpha^{2n-2m}=\frac{\pi}{2}{F_{1}}\left(\tfrac{1}{2};\tfrac{1}{2},1;1;k^{2},\alpha^{2}\right)}}$ \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{n}}{n!}\sum_{m=0}^{n}\frac{\Pochhammersym{\tfrac{1}{2}}{m}}{m!}k^{2m}\alpha^{2n-2m} = \frac{\pi}{2}\AppellF{1}@{\tfrac{1}{2}}{\tfrac{1}{2}}{1}{1}{k^{2}}{\alpha^{2}}		Error	Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], n],(n)!]*Sum[Divide[Pochhammer[Divide[1,2], m],(m)!]*(k)^(2*m)* \[Alpha]^(2*n - 2*m), {m, 0, n}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*AppellF[1, , Divide[1,2], Divide[1,2], 1, 1]*(k)^(2)*\[Alpha]^(2)	Missing Macro Error	Failure	Skip - symbolical successful subtest	Skipped - Because timed out
19.5.E5	${\displaystyle{\displaystyle q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)}}$ q = \exp@{-\pi\ccompellintKk@{k}/\compellintKk@{k}}	${\displaystyle{\displaystyle r=\frac{1}{16}k^{2},0\leq k,k\leq 1}}$	(exp(- Pi*EllipticCK(k)/EllipticK(k))) = exp(- Pi*EllipticCK(k)/EllipticK(k))	(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]) == Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]	Successful	Successful	-	Successful [Tested: 1]
19.5.E7	${\displaystyle{\displaystyle\lambda=(1-\sqrt{k^{\prime}})/(2(1+\sqrt{k^{\prime}}))}}$ \lambda = (1-\sqrt{k^{\prime}})/(2(1+\sqrt{k^{\prime}}))		lambda = (1 -sqrt(sqrt(1 - (k)^(2))))/(2*(1 +sqrt(sqrt(1 - (k)^(2)))))	\[Lambda] == (1 -Sqrt[Sqrt[1 - (k)^(2)]])/(2*(1 +Sqrt[Sqrt[1 - (k)^(2)]]))	Skipped - no semantic math	Skipped - no semantic math	-	-
19.5.E8	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2}}}$ \compellintKk@{k} = \frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2}	${\displaystyle{\displaystyle|q|<1}}$	EllipticK(k) = (Pi)/(2)*(1 + 2*sum((exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))^(2)	EllipticK[(k)^2] == Divide[Pi,2]*((1 + 2*Sum[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None]))^(2)	Failure	Failure	Error	Failed [1 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]}
19.5.E9	${\displaystyle{\displaystyle E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K\left(k\right)}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}}}}$ \compellintEk@{k} = \compellintKk@{k}+\frac{2\pi^{2}}{\compellintKk@{k}}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}}	${\displaystyle{\displaystyle|q|<1}}$	EllipticE(k) = EllipticK(k)+(2*(Pi)^(2))/(EllipticK(k))*(sum((- 1)^(n)* (n)^(2)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))/(1 + 2*sum((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))	EllipticE[(k)^2] == EllipticK[(k)^2]+Divide[2*(Pi)^(2),EllipticK[(k)^2]]*Divide[Sum[(- 1)^(n)* (n)^(2)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None],1 + 2*Sum[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None]]	Failure	Failure	Error	Skipped - Because timed out
19.5.E10	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m})}}$ \compellintKk@{k} = \frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m})		EllipticK(k) = (Pi)/(2)*product(1 + k[m], m = 1..infinity)	EllipticK[(k)^2] == Divide[Pi,2]*Product[1 + Subscript[k, m], {m, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [30 / 30] Result: Plus[DirectedInfinity[], Times[-1.5707963267948966, NProduct[Plus[1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {m, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[k, 1], Rule[Subscript[k, m], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Complex[0.8428751774062981, -1.0782578237498217], Times[-1.5707963267948966, NProduct[Plus[1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {m, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[k, 2], Rule[Subscript[k, m], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.5.E11	${\displaystyle{\displaystyle k_{m+1}=\frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}}}}$ k_{m+1} = \frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}}		k[m + 1] = (1 -sqrt(1 - (k[m])^(2)))/(1 +sqrt(1 - (k[m])^(2)))	Subscript[k, m + 1] == Divide[1 -Sqrt[1 - (Subscript[k, m])^(2)],1 +Sqrt[1 - (Subscript[k, m])^(2)]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.6#Ex1	${\displaystyle{\displaystyle K\left(0\right)=E\left(0\right)}}$ \compellintKk@{0} = \compellintEk@{0}		EllipticK(0) = EllipticE(0)	EllipticK[(0)^2] == EllipticE[(0)^2]	Successful	Successful	-	Successful [Tested: 1]
19.6#Ex1	${\displaystyle{\displaystyle E\left(0\right)={K^{\prime}}\left(1\right)}}$ \compellintEk@{0} = \ccompellintKk@{1}		EllipticE(0) = EllipticCK(1)	EllipticE[(0)^2] == EllipticK[1-(1)^2]	Successful	Successful	-	Successful [Tested: 1]
19.6#Ex1	${\displaystyle{\displaystyle{K^{\prime}}\left(1\right)={E^{\prime}}\left(1\right)}}$ \ccompellintKk@{1} = \ccompellintEk@{1}		EllipticCK(1) = EllipticCE(1)	EllipticK[1-(1)^2] == EllipticE[1-(1)^2]	Successful	Successful	-	Successful [Tested: 1]
19.6#Ex1	${\displaystyle{\displaystyle{E^{\prime}}\left(1\right)=\tfrac{1}{2}\pi}}$ \ccompellintEk@{1} = \tfrac{1}{2}\pi		EllipticCE(1) = (1)/(2)*Pi	EllipticE[1-(1)^2] == Divide[1,2]*Pi	Successful	Successful	-	Successful [Tested: 1]
19.6#Ex2	${\displaystyle{\displaystyle K\left(1\right)={K^{\prime}}\left(0\right)}}$ \compellintKk@{1} = \ccompellintKk@{0}		EllipticK(1) = EllipticCK(0)	EllipticK[(1)^2] == EllipticK[1-(0)^2]	Error	Failure	-	Failed [1 / 1] Result: Indeterminate Test Values: {}
19.6#Ex2	${\displaystyle{\displaystyle{K^{\prime}}\left(0\right)=\infty}}$ \ccompellintKk@{0} = \infty		EllipticCK(0) = infinity	EllipticK[1-(0)^2] == Infinity	Error	Failure	-	Failed [1 / 1] Result: Indeterminate Test Values: {}
19.6#Ex3	${\displaystyle{\displaystyle E\left(1\right)={E^{\prime}}\left(0\right)}}$ \compellintEk@{1} = \ccompellintEk@{0}		EllipticE(1) = EllipticCE(0)	EllipticE[(1)^2] == EllipticE[1-(0)^2]	Successful	Successful	-	Successful [Tested: 1]
19.6#Ex3	${\displaystyle{\displaystyle{E^{\prime}}\left(0\right)=1}}$ \ccompellintEk@{0} = 1		EllipticCE(0) = 1	EllipticE[1-(0)^2] == 1	Successful	Successful	-	Successful [Tested: 1]
19.6#Ex4	${\displaystyle{\displaystyle\Pi\left(k^{2},k\right)=E\left(k\right)/{k^{\prime}}^{2}}}$ \compellintPik@{k^{2}}{k} = \compellintEk@{k}/{k^{\prime}}^{2}	${\displaystyle{\displaystyle k^{2}<1}}$	EllipticPi((k)^(2), k) = EllipticE(k)/(1 - (k)^(2))	EllipticPi[(k)^(2), (k)^2] == EllipticE[(k)^2]/(1 - (k)^(2))	Successful	Successful	-	Successful [Tested: 0]
19.6#Ex5	${\displaystyle{\displaystyle\Pi\left(-k,k\right)=\tfrac{1}{4}\pi(1+k)^{-1}+\tfrac{1}{2}K\left(k\right)}}$ \compellintPik@{-k}{k} = \tfrac{1}{4}\pi(1+k)^{-1}+\tfrac{1}{2}\compellintKk@{k}	${\displaystyle{\displaystyle 0\leq k^{2},k^{2}<1}}$	EllipticPi(- k, k) = (1)/(4)*Pi*(1 + k)^(- 1)+(1)/(2)*EllipticK(k)	EllipticPi[- k, (k)^2] == Divide[1,4]*Pi*(1 + k)^(- 1)+Divide[1,2]*EllipticK[(k)^2]	Failure	Failure	Error	Skip - No test values generated
19.6.E3	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k\right)}}$ \compellintPik@{\alpha^{2}}{0} = \pi/(2\sqrt{1-\alpha^{2}}),\quad\compellintPik@{0}{k}	${\displaystyle{\displaystyle-\infty<\alpha^{2},\alpha^{2}<1}}$	EllipticPi((alpha)^(2), 0) = Pi/(2*sqrt(1 - (alpha)^(2)))	EllipticPi[\[Alpha]^(2), (0)^2] == Pi/(2*Sqrt[1 - \[Alpha]^(2)])	Successful	Failure	Skip - symbolical successful subtest	Error
19.6.E3	${\displaystyle{\displaystyle\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k\right)=K\left(k\right)}}$ \pi/(2\sqrt{1-\alpha^{2}}),\quad\compellintPik@{0}{k} = \compellintKk@{k}	${\displaystyle{\displaystyle-\infty<\alpha^{2},\alpha^{2}<1}}$	Pi/(2*sqrt(1 - (alpha)^(2)))	Pi/(2*Sqrt[1 - \[Alpha]^(2)])	Failure	Failure	Error	Error
19.6.E5	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=K\left(k\right)-\Pi\left(k^{2}/\alpha^{2},k\right)}}$ \compellintPik@{\alpha^{2}}{k} = \compellintKk@{k}-\compellintPik@{k^{2}/\alpha^{2}}{k}		EllipticPi((alpha)^(2), k) = EllipticK(k)- EllipticPi((k)^(2)/(alpha)^(2), k)	EllipticPi[\[Alpha]^(2), (k)^2] == EllipticK[(k)^2]- EllipticPi[(k)^(2)/\[Alpha]^(2), (k)^2]	Failure	Failure	Error	Failed [9 / 9] Result: Indeterminate Test Values: {Rule[k, 1], Rule[α, 1.5]} Result: Complex[-1.593078238683172, 2.4424906541753444*^-15] Test Values: {Rule[k, 2], Rule[α, 1.5]} ... skip entries to safe data
19.6#Ex8	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},0\right)=0}}$ \compellintPik@{\alpha^{2}}{0} = 0		EllipticPi((alpha)^(2), 0) = 0	EllipticPi[\[Alpha]^(2), (0)^2] == 0	Failure	Failure	Failed [3 / 3] Result: -1.404962946*I Test Values: {alpha = 3/2} Result: 1.813799364 Test Values: {alpha = 1/2} ... skip entries to safe data	Failed [3 / 3] Result: Complex[0.0, -1.4049629462081452] Test Values: {Rule[α, 1.5]} Result: 1.813799364234218 Test Values: {Rule[α, 0.5]} ... skip entries to safe data
19.6#Ex11	${\displaystyle{\displaystyle F\left(0,k\right)=0}}$ \incellintFk@{0}{k} = 0		EllipticF(sin(0), k) = 0	EllipticF[0, (k)^2] == 0	Successful	Successful	-	Successful [Tested: 3]
19.6#Ex12	${\displaystyle{\displaystyle F\left(\phi,0\right)=\phi}}$ \incellintFk@{\phi}{0} = \phi		EllipticF(sin(phi), 0) = phi	EllipticF[\[Phi], (0)^2] == \[Phi]	Failure	Successful	Failed [2 / 10] Result: .858407346 Test Values: {phi = -2} Result: -.858407346 Test Values: {phi = 2}	Successful [Tested: 10]
19.6#Ex13	${\displaystyle{\displaystyle F\left(\tfrac{1}{2}\pi,1\right)=\infty}}$ \incellintFk@{\tfrac{1}{2}\pi}{1} = \infty		EllipticF(sin((1)/(2)*Pi), 1) = infinity	EllipticF[Divide[1,2]*Pi, (1)^2] == Infinity	Error	Failure	-	Failed [1 / 1] Result: Indeterminate Test Values: {}
19.6#Ex14	${\displaystyle{\displaystyle F\left(\tfrac{1}{2}\pi,k\right)=K\left(k\right)}}$ \incellintFk@{\tfrac{1}{2}\pi}{k} = \compellintKk@{k}		EllipticF(sin((1)/(2)*Pi), k) = EllipticK(k)	EllipticF[Divide[1,2]*Pi, (k)^2] == EllipticK[(k)^2]	Successful	Successful	-	Successful [Tested: 3]
19.6#Ex15	${\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{F\left(\phi,k\right)}{\phi}=1}}$ \lim_{\phi\to 0}\ifrac{\incellintFk@{\phi}{k}}{\phi} = 1		limit((EllipticF(sin(phi), k))/(phi), phi = 0) = 1	Limit[Divide[EllipticF[\[Phi], (k)^2],\[Phi]], \[Phi] -> 0, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 3]
19.6.E8	${\displaystyle{\displaystyle F\left(\phi,1\right)=(\sin\phi)R_{C}\left(1,{\cos^{2}}\phi\right)}}$ \incellintFk@{\phi}{1} = (\sin@@{\phi})\CarlsonellintRC@{1}{\cos^{2}@@{\phi}}		Error	EllipticF[\[Phi], (1)^2] == (Sin[\[Phi]])*1/Sqrt[(Cos[\[Phi]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(1)/((Cos[\[Phi]])^(2))]	Missing Macro Error	Failure	-	Failed [2 / 10] Result: DirectedInfinity[] Test Values: {Rule[ϕ, -2]} Result: DirectedInfinity[] Test Values: {Rule[ϕ, 2]}
19.6.E8	${\displaystyle{\displaystyle(\sin\phi)R_{C}\left(1,{\cos^{2}}\phi\right)={\operatorname{gd}^{-1}}\left(\phi\right)}}$ (\sin@@{\phi})\CarlsonellintRC@{1}{\cos^{2}@@{\phi}} = \aGudermannian@{\phi}	${\displaystyle{\displaystyle-\frac{1}{2}\pi<(\phi),(\phi)<\frac{1}{2}\pi}}$	Error	(Sin[\[Phi]])*1/Sqrt[(Cos[\[Phi]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(1)/((Cos[\[Phi]])^(2))] == InverseGudermannian[\[Phi]]	Missing Macro Error	Failure	-	Successful [Tested: 4]
19.6#Ex16	${\displaystyle{\displaystyle E\left(0,k\right)=0}}$ \incellintEk@{0}{k} = 0		EllipticE(sin(0), k) = 0	EllipticE[0, (k)^2] == 0	Successful	Successful	-	Successful [Tested: 3]
19.6#Ex17	${\displaystyle{\displaystyle E\left(\phi,0\right)=\phi}}$ \incellintEk@{\phi}{0} = \phi		EllipticE(sin(phi), 0) = phi	EllipticE[\[Phi], (0)^2] == \[Phi]	Failure	Successful	Failed [2 / 10] Result: .858407346 Test Values: {phi = -2} Result: -.858407346 Test Values: {phi = 2}	Successful [Tested: 10]
19.6#Ex18	${\displaystyle{\displaystyle E\left(\tfrac{1}{2}\pi,1\right)=1}}$ \incellintEk@{\tfrac{1}{2}\pi}{1} = 1		EllipticE(sin((1)/(2)*Pi), 1) = 1	EllipticE[Divide[1,2]*Pi, (1)^2] == 1	Successful	Successful	-	Successful [Tested: 1]
19.6#Ex19	${\displaystyle{\displaystyle E\left(\phi,1\right)=\sin\phi}}$ \incellintEk@{\phi}{1} = \sin@@{\phi}		EllipticE(sin(phi), 1) = sin(phi)	EllipticE[\[Phi], (1)^2] == Sin[\[Phi]]	Successful	Failure	-	Failed [2 / 10] Result: -0.1814051463486368 Test Values: {Rule[ϕ, -2]} Result: 0.1814051463486368 Test Values: {Rule[ϕ, 2]}
19.6#Ex20	${\displaystyle{\displaystyle E\left(\tfrac{1}{2}\pi,k\right)=E\left(k\right)}}$ \incellintEk@{\tfrac{1}{2}\pi}{k} = \compellintEk@{k}		EllipticE(sin((1)/(2)*Pi), k) = EllipticE(k)	EllipticE[Divide[1,2]*Pi, (k)^2] == EllipticE[(k)^2]	Successful	Successful	-	Successful [Tested: 3]
19.6.E10	${\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{E\left(\phi,k\right)}{\phi}=1}}$ \lim_{\phi\to 0}\ifrac{\incellintEk@{\phi}{k}}{\phi} = 1		limit((EllipticE(sin(phi), k))/(phi), phi = 0) = 1	Limit[Divide[EllipticE[\[Phi], (k)^2],\[Phi]], \[Phi] -> 0, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 3]
19.6#Ex21	${\displaystyle{\displaystyle\Pi\left(0,\alpha^{2},k\right)=0}}$ \incellintPik@{0}{\alpha^{2}}{k} = 0		EllipticPi(sin(0), (alpha)^(2), k) = 0	EllipticPi[\[Alpha]^(2), 0,(k)^2] == 0	Successful	Successful	-	Successful [Tested: 9]
19.6#Ex22	${\displaystyle{\displaystyle\Pi\left(\phi,0,0\right)=\phi}}$ \incellintPik@{\phi}{0}{0} = \phi		EllipticPi(sin(phi), 0, 0) = phi	EllipticPi[0, \[Phi],(0)^2] == \[Phi]	Failure	Successful	Failed [2 / 10] Result: .858407346 Test Values: {phi = -2} Result: -.858407346 Test Values: {phi = 2}	Successful [Tested: 10]
19.6#Ex23	${\displaystyle{\displaystyle\Pi\left(\phi,1,0\right)=\tan\phi}}$ \incellintPik@{\phi}{1}{0} = \tan@@{\phi}		EllipticPi(sin(phi), 1, 0) = tan(phi)	EllipticPi[1, \[Phi],(0)^2] == Tan[\[Phi]]	Failure	Successful	Failed [2 / 10] Result: -4.370079726 Test Values: {phi = -2} Result: 4.370079726 Test Values: {phi = 2}	Successful [Tested: 10]
19.6#Ex24	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},0\right)=R_{C}\left(c-1,c-\alpha^{2}\right)}}$ \incellintPik@{\phi}{\alpha^{2}}{0} = \CarlsonellintRC@{c-1}{c-\alpha^{2}}		Error	EllipticPi[\[Alpha]^(2), \[Phi],(0)^2] == 1/Sqrt[c - \[Alpha]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(c - 1)/(c - \[Alpha]^(2))]	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[0.4032669574270382, 0.8997227991212673] Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.17167863497284278, 0.9673069947694621] Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.6#Ex25	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},1\right)=\frac{1}{1-\alpha^{2}}\left(R_{C}\left(c,c-1\right)-\alpha^{2}R_{C}\left(c,c-\alpha^{2}\right)\right)}}$ \incellintPik@{\phi}{\alpha^{2}}{1} = \frac{1}{1-\alpha^{2}}\left(\CarlsonellintRC@{c}{c-1}-\alpha^{2}\CarlsonellintRC@{c}{c-\alpha^{2}}\right)		Error	EllipticPi[\[Alpha]^(2), \[Phi],(1)^2] == Divide[1,1 - \[Alpha]^(2)]*(1/Sqrt[c - 1]*Hypergeometric2F1[1/2,1/2,3/2,1-(c)/(c - 1)]- \[Alpha]^(2)* 1/Sqrt[c - \[Alpha]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(c)/(c - \[Alpha]^(2))])	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[0.39392267303966433, 0.8870442763896845] Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.15564928813724596, 0.9274825692848638] Test Values: {Rule[c, -1.5], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.6#Ex26	${\displaystyle{\displaystyle\Pi\left(\phi,1,1\right)=\tfrac{1}{2}(R_{C}\left(c,c-1\right)+\sqrt{c}(c-1)^{-1})}}$ \incellintPik@{\phi}{1}{1} = \tfrac{1}{2}(\CarlsonellintRC@{c}{c-1}+\sqrt{c}(c-1)^{-1})		Error	EllipticPi[1, \[Phi],(1)^2] == Divide[1,2]*(1/Sqrt[c - 1]*Hypergeometric2F1[1/2,1/2,3/2,1-(c)/(c - 1)]+Sqrt[c]*(c - 1)^(- 1))	Missing Macro Error	Failure	-	Failed [60 / 60] Result: Complex[0.42461599644771203, 0.9033982135739806] Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.19222674503116347, 1.0138365568937844] Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.6#Ex27	${\displaystyle{\displaystyle\Pi\left(\phi,0,k\right)=F\left(\phi,k\right)}}$ \incellintPik@{\phi}{0}{k} = \incellintFk@{\phi}{k}		EllipticPi(sin(phi), 0, k) = EllipticF(sin(phi), k)	EllipticPi[0, \[Phi],(k)^2] == EllipticF[\[Phi], (k)^2]	Successful	Successful	-	Successful [Tested: 30]
19.6#Ex28	${\displaystyle{\displaystyle\Pi\left(\phi,k^{2},k\right)=\frac{1}{{k^{\prime}}^{2}}\left(E\left(\phi,k\right)-\frac{k^{2}}{\Delta}\sin\phi\cos\phi\right)}}$ \incellintPik@{\phi}{k^{2}}{k} = \frac{1}{{k^{\prime}}^{2}}\left(\incellintEk@{\phi}{k}-\frac{k^{2}}{\Delta}\sin@@{\phi}\cos@@{\phi}\right)		EllipticPi(sin(phi), (k)^(2), k) = (1)/(1 - (k)^(2))*(EllipticE(sin(phi), k)-((k)^(2))/(Delta)*sin(phi)*cos(phi))	EllipticPi[(k)^(2), \[Phi],(k)^2] == Divide[1,1 - (k)^(2)]*(EllipticE[\[Phi], (k)^2]-Divide[(k)^(2),\[CapitalDelta]]*Sin[\[Phi]]*Cos[\[Phi]])	Failure	Failure	Failed [300 / 300] Result: Float(infinity)+Float(infinity)*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.4574406724+1.116997071*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Indeterminate Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.8161437733664769, 0.6845645198965172] Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.6#Ex29	${\displaystyle{\displaystyle\Pi\left(\phi,1,k\right)=F\left(\phi,k\right)-\frac{1}{{k^{\prime}}^{2}}(E\left(\phi,k\right)-\Delta\tan\phi)}}$ \incellintPik@{\phi}{1}{k} = \incellintFk@{\phi}{k}-\frac{1}{{k^{\prime}}^{2}}(\incellintEk@{\phi}{k}-\Delta\tan@@{\phi})		EllipticPi(sin(phi), 1, k) = EllipticF(sin(phi), k)-(1)/(1 - (k)^(2))*(EllipticE(sin(phi), k)- Delta*tan(phi))	EllipticPi[1, \[Phi],(k)^2] == EllipticF[\[Phi], (k)^2]-Divide[1,1 - (k)^(2)]*(EllipticE[\[Phi], (k)^2]- \[CapitalDelta]*Tan[\[Phi]])	Failure	Failure	Failed [300 / 300] Result: Float(infinity)+Float(infinity)*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.5381374542+.4861981155*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Indeterminate Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.12805668293605252, 0.0652384492706456] Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.6#Ex30	${\displaystyle{\displaystyle\Pi\left(\tfrac{1}{2}\pi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)}}$ \incellintPik@{\tfrac{1}{2}\pi}{\alpha^{2}}{k} = \compellintPik@{\alpha^{2}}{k}		EllipticPi(sin((1)/(2)*Pi), (alpha)^(2), k) = EllipticPi((alpha)^(2), k)	EllipticPi[\[Alpha]^(2), Divide[1,2]*Pi,(k)^2] == EllipticPi[\[Alpha]^(2), (k)^2]	Successful	Successful	-	Successful [Tested: 9]
19.6#Ex31	${\displaystyle{\displaystyle\lim_{\phi\to 0}\ifrac{\Pi\left(\phi,\alpha^{2},k\right)}{\phi}=1}}$ \lim_{\phi\to 0}\ifrac{\incellintPik@{\phi}{\alpha^{2}}{k}}{\phi} = 1		limit((EllipticPi(sin(phi), (alpha)^(2), k))/(phi), phi = 0) = 1	Limit[Divide[EllipticPi[\[Alpha]^(2), \[Phi],(k)^2],\[Phi]], \[Phi] -> 0, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 9]
19.6#Ex32	${\displaystyle{\displaystyle R_{C}\left(x,x\right)=x^{-1/2}}}$ \CarlsonellintRC@{x}{x} = x^{-1/2}		Error	1/Sqrt[x]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(x)] == (x)^(- 1/2)	Missing Macro Error	Successful	-	Successful [Tested: 3]
19.6#Ex33	${\displaystyle{\displaystyle R_{C}\left(\lambda x,\lambda y\right)=\lambda^{-1/2}R_{C}\left(x,y\right)}}$ \CarlsonellintRC@{\lambda x}{\lambda y} = \lambda^{-1/2}\CarlsonellintRC@{x}{y}		Error	1/Sqrt[\[Lambda]*y]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Lambda]*x)/(\[Lambda]*y)] == \[Lambda]^(- 1/2)* 1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]	Missing Macro Error	Failure	-	Failed [75 / 180] Result: Complex[2.0541315094196904, 2.1051836996148214] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[2.941079989400646, 0.036099349881403064] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.6#Ex35	${\displaystyle{\displaystyle R_{C}\left(0,y\right)=\tfrac{1}{2}\pi y^{-1/2}}}$ \CarlsonellintRC@{0}{y} = \tfrac{1}{2}\pi y^{-1/2}	${\displaystyle{\displaystyle|\operatorname{ph}y|<\pi}}$	Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(0)/(y)] == Divide[1,2]*Pi*(y)^(- 1/2)	Missing Macro Error	Successful	-	Successful [Tested: 3]
19.6#Ex36	${\displaystyle{\displaystyle R_{C}\left(0,y\right)=0}}$ \CarlsonellintRC@{0}{y} = 0	${\displaystyle{\displaystyle y<0}}$	Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(0)/(y)] == 0	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Complex[0.0, -1.2825498301618643] Test Values: {Rule[y, -1.5]} Result: Complex[0.0, -2.221441469079183] Test Values: {Rule[y, -0.5]} ... skip entries to safe data
19.7.E1	${\displaystyle{\displaystyle E\left(k\right){K^{\prime}}\left(k\right)+{E^{\prime}}\left(k\right)K\left(k\right)-K\left(k\right){K^{\prime}}\left(k\right)=\tfrac{1}{2}\pi}}$ \compellintEk@{k}\ccompellintKk@{k}+\ccompellintEk@{k}\compellintKk@{k}-\compellintKk@{k}\ccompellintKk@{k} = \tfrac{1}{2}\pi		EllipticE(k)*EllipticCK(k)+ EllipticCE(k)*EllipticK(k)- EllipticK(k)*EllipticCK(k) = (1)/(2)*Pi	EllipticE[(k)^2]*EllipticK[1-(k)^2]+ EllipticE[1-(k)^2]*EllipticK[(k)^2]- EllipticK[(k)^2]*EllipticK[1-(k)^2] == Divide[1,2]*Pi	Failure	Failure	Error	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.7#Ex1	${\displaystyle{\displaystyle K\left(ik/k^{\prime}\right)=k^{\prime}K\left(k\right)}}$ \compellintKk@{ik/k^{\prime}} = k^{\prime}\compellintKk@{k}		EllipticK(I*k/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*EllipticK(k)	EllipticK[(I*k/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*EllipticK[(k)^2]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.220446049250313*^-16, -2.9198052634126777] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex2	${\displaystyle{\displaystyle K\left(-ik^{\prime}/k\right)=kK\left(k^{\prime}\right)}}$ \compellintKk@{-ik^{\prime}/k} = k\compellintKk@{k^{\prime}}		EllipticK(- I*sqrt(1 - (k)^(2))/k) = k*EllipticK(sqrt(1 - (k)^(2)))	EllipticK[(- I*Sqrt[1 - (k)^(2)]/k)^2] == k*EllipticK[(Sqrt[1 - (k)^(2)])^2]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
19.7#Ex3	${\displaystyle{\displaystyle E\left(ik/k^{\prime}\right)=(1/k^{\prime})E\left(k\right)}}$ \compellintEk@{ik/k^{\prime}} = (1/k^{\prime})\compellintEk@{k}		EllipticE(I*k/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*EllipticE(k)	EllipticE[(I*k/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*EllipticE[(k)^2]	Failure	Failure	Failed [3 / 3] Result: Float(infinity)+Float(infinity)*I Test Values: {k = 1} Result: .6e-9+.4691535424*I Test Values: {k = 2} ... skip entries to safe data	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-5.551115123125783*^-16, 0.46915354293820644] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex4	${\displaystyle{\displaystyle E\left(-ik^{\prime}/k\right)=(1/k)E\left(k^{\prime}\right)}}$ \compellintEk@{-ik^{\prime}/k} = (1/k)\compellintEk@{k^{\prime}}		EllipticE(- I*sqrt(1 - (k)^(2))/k) = (1/k)*EllipticE(sqrt(1 - (k)^(2)))	EllipticE[(- I*Sqrt[1 - (k)^(2)]/k)^2] == (1/k)*EllipticE[(Sqrt[1 - (k)^(2)])^2]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
19.7#Ex5	${\displaystyle{\displaystyle K\left(1/k\right)=k(K\left(k\right)-\mathrm{i}K\left(k^{\prime}\right))}}$ \compellintKk@{1/k} = k(\compellintKk@{k}-\iunit\compellintKk@{k^{\prime}})		EllipticK(1/k) = k*(EllipticK(k)- I*EllipticK(sqrt(1 - (k)^(2))))	EllipticK[(1/k)^2] == k*(EllipticK[(k)^2]- I*EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.220446049250313*^-16, 4.313031294999287] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex5	${\displaystyle{\displaystyle K\left(1/k\right)=k(K\left(k\right)+\mathrm{i}K\left(k^{\prime}\right))}}$ \compellintKk@{1/k} = k(\compellintKk@{k}+\iunit\compellintKk@{k^{\prime}})		EllipticK(1/k) = k*(EllipticK(k)+ I*EllipticK(sqrt(1 - (k)^(2))))	EllipticK[(1/k)^2] == k*(EllipticK[(k)^2]+ I*EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.7#Ex6	${\displaystyle{\displaystyle K\left(1/k^{\prime}\right)=k^{\prime}(K\left(k^{\prime}\right)+\mathrm{i}K\left(k\right))}}$ \compellintKk@{1/k^{\prime}} = k^{\prime}(\compellintKk@{k^{\prime}}+\iunit\compellintKk@{k})		EllipticK(1/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*(EllipticK(sqrt(1 - (k)^(2)))+ I*EllipticK(k))	EllipticK[(1/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*(EllipticK[(Sqrt[1 - (k)^(2)])^2]+ I*EllipticK[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[2.9198052634126785, -3.7351946687866775] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex6	${\displaystyle{\displaystyle K\left(1/k^{\prime}\right)=k^{\prime}(K\left(k^{\prime}\right)-\mathrm{i}K\left(k\right))}}$ \compellintKk@{1/k^{\prime}} = k^{\prime}(\compellintKk@{k^{\prime}}-\iunit\compellintKk@{k})		EllipticK(1/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*(EllipticK(sqrt(1 - (k)^(2)))- I*EllipticK(k))	EllipticK[(1/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*(EllipticK[(Sqrt[1 - (k)^(2)])^2]- I*EllipticK[(k)^2])	Failure	Failure	Error	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.7#Ex7	${\displaystyle{\displaystyle E\left(1/k\right)=(1/k)\left(E\left(k\right)+\mathrm{i}E\left(k^{\prime}\right)-{k^{\prime}}^{2}K\left(k\right)-\mathrm{i}k^{2}K\left(k^{\prime}\right)\right)}}$ \compellintEk@{1/k} = (1/k)\left(\compellintEk@{k}+\iunit\compellintEk@{k^{\prime}}-{k^{\prime}}^{2}\compellintKk@{k}-\iunit k^{2}\compellintKk@{k^{\prime}}\right)		EllipticE(1/k) = (1/k)*(EllipticE(k)+ I*EllipticE(sqrt(1 - (k)^(2)))-1 - (k)^(2)*EllipticK(k)- I*(k)^(2)* EllipticK(sqrt(1 - (k)^(2))))	EllipticE[(1/k)^2] == (1/k)*(EllipticE[(k)^2]+ I*EllipticE[(Sqrt[1 - (k)^(2)])^2]-1 - (k)^(2)*EllipticK[(k)^2]- I*(k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: Complex[3.4500631209220436, -1.8829831432620088] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex7	${\displaystyle{\displaystyle E\left(1/k\right)=(1/k)\left(E\left(k\right)-\mathrm{i}E\left(k^{\prime}\right)-{k^{\prime}}^{2}K\left(k\right)+\mathrm{i}k^{2}K\left(k^{\prime}\right)\right)}}$ \compellintEk@{1/k} = (1/k)\left(\compellintEk@{k}-\iunit\compellintEk@{k^{\prime}}-{k^{\prime}}^{2}\compellintKk@{k}+\iunit k^{2}\compellintKk@{k^{\prime}}\right)		EllipticE(1/k) = (1/k)*(EllipticE(k)- I*EllipticE(sqrt(1 - (k)^(2)))-1 - (k)^(2)*EllipticK(k)+ I*(k)^(2)* EllipticK(sqrt(1 - (k)^(2))))	EllipticE[(1/k)^2] == (1/k)*(EllipticE[(k)^2]- I*EllipticE[(Sqrt[1 - (k)^(2)])^2]-1 - (k)^(2)*EllipticK[(k)^2]+ I*(k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: Complex[3.4500631209220436, -3.773902383124376] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex8	${\displaystyle{\displaystyle E\left(1/k^{\prime}\right)=(1/k^{\prime})\left(E\left(k^{\prime}\right)-\mathrm{i}E\left(k\right)-k^{2}K\left(k^{\prime}\right)+\mathrm{i}{k^{\prime}}^{2}K\left(k\right)\right)}}$ \compellintEk@{1/k^{\prime}} = (1/k^{\prime})\left(\compellintEk@{k^{\prime}}-\iunit\compellintEk@{k}-k^{2}\compellintKk@{k^{\prime}}+\iunit{k^{\prime}}^{2}\compellintKk@{k}\right)		EllipticE(1/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*(EllipticE(sqrt(1 - (k)^(2)))- I*EllipticE(k)- (k)^(2)* EllipticK(sqrt(1 - (k)^(2)))+ I*1 - (k)^(2)*EllipticK(k))	EllipticE[(1/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*(EllipticE[(Sqrt[1 - (k)^(2)])^2]- I*EllipticE[(k)^2]- (k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2]+ I*1 - (k)^(2)*EllipticK[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-1.1384238737361991, -2.262384972182541] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex8	${\displaystyle{\displaystyle E\left(1/k^{\prime}\right)=(1/k^{\prime})\left(E\left(k^{\prime}\right)+\mathrm{i}E\left(k\right)-k^{2}K\left(k^{\prime}\right)-\mathrm{i}{k^{\prime}}^{2}K\left(k\right)\right)}}$ \compellintEk@{1/k^{\prime}} = (1/k^{\prime})\left(\compellintEk@{k^{\prime}}+\iunit\compellintEk@{k}-k^{2}\compellintKk@{k^{\prime}}-\iunit{k^{\prime}}^{2}\compellintKk@{k}\right)		EllipticE(1/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*(EllipticE(sqrt(1 - (k)^(2)))+ I*EllipticE(k)- (k)^(2)* EllipticK(sqrt(1 - (k)^(2)))- I*1 - (k)^(2)*EllipticK(k))	EllipticE[(1/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*(EllipticE[(Sqrt[1 - (k)^(2)])^2]+ I*EllipticE[(k)^2]- (k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2]- I*1 - (k)^(2)*EllipticK[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-0.45287687829515355, -3.814134176668458] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex19	${\displaystyle{\displaystyle F\left(i\phi,k\right)=iF\left(\psi,k^{\prime}\right)}}$ \incellintFk@{i\phi}{k} = i\incellintFk@{\psi}{k^{\prime}}		EllipticF(sin(I*phi), k) = I*EllipticF(sin(psi), sqrt(1 - (k)^(2)))	EllipticF[I*\[Phi], (k)^2] == I*EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]	Failure	Failure	Failed [300 / 300] Result: .1428695990-.263545696e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1} Result: .749290340e-1-.334629029e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.020142137049999537, -0.0010462389457662757] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.015860617706546204, -0.003938067237051424] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.7#Ex20	${\displaystyle{\displaystyle E\left(i\phi,k\right)=i\left(F\left(\psi,k^{\prime}\right)-E\left(\psi,k^{\prime}\right)+(\tan\psi)\sqrt{1-{k^{\prime}}^{2}{\sin^{2}}\psi}\right)}}$ \incellintEk@{i\phi}{k} = i\left(\incellintFk@{\psi}{k^{\prime}}-\incellintEk@{\psi}{k^{\prime}}+(\tan@@{\psi})\sqrt{1-{k^{\prime}}^{2}\sin^{2}@@{\psi}}\right)		EllipticE(sin(I*phi), k) = I*(EllipticF(sin(psi), sqrt(1 - (k)^(2)))- EllipticE(sin(psi), sqrt(1 - (k)^(2)))+(tan(psi))*sqrt(1 -1 - (k)^(2)*(sin(psi))^(2)))	EllipticE[I*\[Phi], (k)^2] == I*(EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]- EllipticE[\[Psi], (Sqrt[1 - (k)^(2)])^2]+(Tan[\[Psi]])*Sqrt[1 -1 - (k)^(2)*(Sin[\[Psi]])^(2)])	Failure	Failure	Failed [300 / 300] Result: -.9970133474-.1125517221*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -2.257467281-.7782721018*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.3893501368763376, 0.20738614458301174] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.6710974690872284, 0.0060773305020283] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.7#Ex21	${\displaystyle{\displaystyle\Pi\left(i\phi,\alpha^{2},k\right)=i\left(F\left(\psi,k^{\prime}\right)-\alpha^{2}\Pi\left(\psi,1-\alpha^{2},k^{\prime}\right)\right)/{(1-\alpha^{2})}}}$ \incellintPik@{i\phi}{\alpha^{2}}{k} = i\left(\incellintFk@{\psi}{k^{\prime}}-\alpha^{2}\incellintPik@{\psi}{1-\alpha^{2}}{k^{\prime}}\right)/{(1-\alpha^{2})}		EllipticPi(sin(I*phi), (alpha)^(2), k) = I*(EllipticF(sin(psi), sqrt(1 - (k)^(2)))- (alpha)^(2)* EllipticPi(sin(psi), 1 - (alpha)^(2), sqrt(1 - (k)^(2))))/(1 - (alpha)^(2))	EllipticPi[\[Alpha]^(2), I*\[Phi],(k)^2] == I*(EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]- \[Alpha]^(2)* EllipticPi[1 - \[Alpha]^(2), \[Psi],(Sqrt[1 - (k)^(2)])^2])/(1 - \[Alpha]^(2))	Failure	Failure	Failed [292 / 300] Result: .926834363e-2-.484444094e-1*I Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.130749569e-2-.277524276e-1*I Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [298 / 300] Result: Complex[0.013291772923717082, -0.006719909387202905] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.00953602334602252, -0.007394575555177196] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex1	${\displaystyle{\displaystyle a_{n+1}=\frac{a_{n}+g_{n}}{2}}}$ a_{n+1} = \frac{a_{n}+g_{n}}{2}		a[n + 1] = (a[n]+ g[n])/(2)	Subscript[a, n + 1] == Divide[Subscript[a, n]+ Subscript[g, n],2]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex2	${\displaystyle{\displaystyle g_{n+1}=\sqrt{a_{n}g_{n}}}}$ g_{n+1} = \sqrt{a_{n}g_{n}}		g[n + 1] = sqrt(a[n]*g[n])	Subscript[g, n + 1] == Sqrt[Subscript[a, n]*Subscript[g, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E2	${\displaystyle{\displaystyle c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}}}$ c_{n} = \sqrt{a_{n}^{2}-g_{n}^{2}}		c[n] = sqrt((a[n])^(2)- (g[n])^(2))	Subscript[c, n] == Sqrt[(Subscript[a, n])^(2)- (Subscript[g, n])^(2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E3	${\displaystyle{\displaystyle c_{n+1}=\frac{a_{n}-g_{n}}{2}}}$ c_{n+1} = \frac{a_{n}-g_{n}}{2}		c[n + 1] = (a[n]- g[n])/(2)	Subscript[c, n + 1] == Divide[Subscript[a, n]- Subscript[g, n],2]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E4	${\displaystyle{\displaystyle\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^{2}{\sin^{2}}\theta}}}}$ \frac{1}{\AGM@{a_{0}}{g_{0}}} = \frac{2}{\pi}\int_{0}^{\pi/2}\frac{\diff{\theta}}{\sqrt{a_{0}^{2}\cos^{2}@@{\theta}+g_{0}^{2}\sin^{2}@@{\theta}}}		(1)/(GaussAGM(a[0], g[0])) = (2)/(Pi)*int((1)/(sqrt((a[0])^(2)*(cos(theta))^(2)+ (g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/2)	Error	Failure	Missing Macro Error	Error	Skip - symbolical successful subtest
19.8.E4	${\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^{2}{\sin^{2}}\theta}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}}}$ \frac{2}{\pi}\int_{0}^{\pi/2}\frac{\diff{\theta}}{\sqrt{a_{0}^{2}\cos^{2}@@{\theta}+g_{0}^{2}\sin^{2}@@{\theta}}} = \frac{1}{\pi}\int_{0}^{\infty}\frac{\diff{t}}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}		(2)/(Pi)*int((1)/(sqrt((a[0])^(2)*(cos(theta))^(2)+ (g[0])^(2)*(sin(theta))^(2))), theta = 0..Pi/2) = (1)/(Pi)*int((1)/(sqrt(t*(t + (a[0])^(2))*(t + (g[0])^(2)))), t = 0..infinity)	Divide[2,Pi]*Integrate[Divide[1,Sqrt[(Subscript[a, 0])^(2)*(Cos[\[Theta]])^(2)+ (Subscript[g, 0])^(2)*(Sin[\[Theta]])^(2)]], {\[Theta], 0, Pi/2}, GenerateConditions->None] == Divide[1,Pi]*Integrate[Divide[1,Sqrt[t*(t + (Subscript[a, 0])^(2))*(t + (Subscript[g, 0])^(2))]], {t, 0, Infinity}, GenerateConditions->None]	Aborted	Aborted	Skipped - Because timed out	Skipped - Because timed out
19.8.E5	${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)}}}$ \compellintKk@{k} = \frac{\pi}{2\AGM@{1}{k^{\prime}}}	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1}}$	EllipticK(k) = (Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))	Error	Failure	Missing Macro Error	Error	-
19.8.E6	${\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)}}$ \compellintEk@{k} = \frac{\pi}{2\AGM@{1}{k^{\prime}}}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,a_{0}=1,g_{0}=k^{\prime}}}$	EllipticE(k) = (Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))*((a[0])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 0..infinity))	Error	Failure	Missing Macro Error	Error	-
19.8.E6	${\displaystyle{\displaystyle\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k\right)\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)}}$ \frac{\pi}{2\AGM@{1}{k^{\prime}}}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right) = \compellintKk@{k}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right)	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,a_{0}=1,g_{0}=k^{\prime}}}$	(Pi)/(2*GaussAGM(1, sqrt(1 - (k)^(2))))*((a[0])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 0..infinity)) = EllipticK(k)*((a[1])^(2)- sum((2)^(n - 1)* (c[n])^(2), n = 2..infinity))	Error	Failure	Missing Macro Error	Error	-
19.8.E7	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right)}}$ \compellintPik@{\alpha^{2}}{k} = \frac{\pi}{4\AGM@{1}{k^{\prime}}}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right)	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,-\infty<\alpha^{2},\alpha^{2}<1}}$	EllipticPi((alpha)^(2), k) = (Pi)/(4*GaussAGM(1, sqrt(1 - (k)^(2))))*(2 +((alpha)^(2))/(1 - (alpha)^(2))*sum(Q[n], n = 0..infinity))	Error	Failure	Missing Macro Error	Error	-
19.8#Ex3	${\displaystyle{\displaystyle p_{n+1}=\frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}}}$ p_{n+1} = \frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}}		p[n + 1] = ((p[n])^(2)+ a[n]*g[n])/(2*p[n])	Subscript[p, n + 1] == Divide[(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n],2*Subscript[p, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex4	${\displaystyle{\displaystyle\varepsilon_{n}=\frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}}}}$ \varepsilon_{n} = \frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}}		varepsilon[n] = ((p[n])^(2)- a[n]*g[n])/((p[n])^(2)+ a[n]*g[n])	Subscript[\[CurlyEpsilon], n] == Divide[(Subscript[p, n])^(2)- Subscript[a, n]*Subscript[g, n],(Subscript[p, n])^(2)+ Subscript[a, n]*Subscript[g, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex5	${\displaystyle{\displaystyle Q_{n+1}=\tfrac{1}{2}Q_{n}\varepsilon_{n}}}$ Q_{n+1} = \tfrac{1}{2}Q_{n}\varepsilon_{n}		Q[n + 1] = (1)/(2)*Q[n]*varepsilon[n]	Subscript[Q, n + 1] == Divide[1,2]*Subscript[Q, n]*Subscript[\[CurlyEpsilon], n]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8.E9	${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}}}$ \compellintPik@{\alpha^{2}}{k} = \frac{\pi}{4\AGM@{1}{k^{\prime}}}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}	${\displaystyle{\displaystyle-\infty<k^{2},k^{2}<1,1<\alpha^{2},\alpha^{2}<\infty}}$	EllipticPi((alpha)^(2), k) = (Pi)/(4*GaussAGM(1, sqrt(1 - (k)^(2))))*((k)^(2))/((k)^(2)- (alpha)^(2))*sum(Q[n], n = 0..infinity)	Error	Failure	Missing Macro Error	Error	-
19.8.E10	${\displaystyle{\displaystyle p_{0}^{2}=1-(k^{2}/\alpha^{2})}}$ p_{0}^{2} = 1-(k^{2}/\alpha^{2})		(p[0])^(2) = 1 -((k)^(2)/(alpha)^(2))	(Subscript[p, 0])^(2) == 1 -((k)^(2)/\[Alpha]^(2))	Skipped - no semantic math	Skipped - no semantic math	-	-
19.8#Ex8	${\displaystyle{\displaystyle K\left(k\right)=(1+k_{1})K\left(k_{1}\right)}}$ \compellintKk@{k} = (1+k_{1})\compellintKk@{k_{1}}		EllipticK(k) = (1 + k[1])*EllipticK(k[1])	EllipticK[(k)^2] == (1 + Subscript[k, 1])*EllipticK[(Subscript[k, 1])^2]	Failure	Failure	Error	Failed [30 / 30] Result: DirectedInfinity[] Test Values: {Rule[k, 1], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.44075376931664, -1.6191557371087932] Test Values: {Rule[k, 2], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex9	${\displaystyle{\displaystyle E\left(k\right)=(1+k^{\prime})E\left(k_{1}\right)-k^{\prime}K\left(k\right)}}$ \compellintEk@{k} = (1+k^{\prime})\compellintEk@{k_{1}}-k^{\prime}\compellintKk@{k}		EllipticE(k) = (1 +sqrt(1 - (k)^(2)))*EllipticE(k[1])-sqrt(1 - (k)^(2))*EllipticK(k)	EllipticE[(k)^2] == (1 +Sqrt[1 - (k)^(2)])*EllipticE[(Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticK[(k)^2]	Failure	Failure	Error	Failed [30 / 30] Result: Indeterminate Test Values: {Rule[k, 1], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.595329372049606, 0.2521613076710463] Test Values: {Rule[k, 2], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex10	${\displaystyle{\displaystyle F\left(\phi,k\right)=\tfrac{1}{2}(1+k_{1})F\left(\phi_{1},k_{1}\right)}}$ \incellintFk@{\phi}{k} = \tfrac{1}{2}(1+k_{1})\incellintFk@{\phi_{1}}{k_{1}}		EllipticF(sin(phi), k) = (1)/(2)*(1 + k[1])*EllipticF(sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi))), k[1])	EllipticF[\[Phi], (k)^2] == Divide[1,2]*(1 + Subscript[k, 1])*EllipticF[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]], (Subscript[k, 1])^2]	Failure	Failure	Failed [300 / 300] Result: .2591790565-.226164263e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: .8581261265-.11942686e-2*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.15619877563526813, 0.03685530383845256] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.6672885103059906, -0.24203301849204312] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex11	${\displaystyle{\displaystyle E\left(\phi,k\right)=\tfrac{1}{2}(1+k^{\prime})E\left(\phi_{1},k_{1}\right)-k^{\prime}F\left(\phi,k\right)+\tfrac{1}{2}(1-k^{\prime})\sin\phi_{1}}}$ \incellintEk@{\phi}{k} = \tfrac{1}{2}(1+k^{\prime})\incellintEk@{\phi_{1}}{k_{1}}-k^{\prime}\incellintFk@{\phi}{k}+\tfrac{1}{2}(1-k^{\prime})\sin@@{\phi_{1}}		EllipticE(sin(phi), k) = (1)/(2)*(1 +sqrt(1 - (k)^(2)))*EllipticE(sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi))), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1)/(2)*(1 -sqrt(1 - (k)^(2)))*sin(phi + arctan(sqrt(1 - (k)^(2))*tan(phi)))	EllipticE[\[Phi], (k)^2] == Divide[1,2]*(1 +Sqrt[1 - (k)^(2)])*EllipticE[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]], (Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticF[\[Phi], (k)^2]+Divide[1,2]*(1 -Sqrt[1 - (k)^(2)])*Sin[\[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]]]	Failure	Failure	Failed [300 / 300] Result: -.627821156e-1-.413169945e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: .886069620e-1-.4575597e-3*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.0022565574667213206, -0.009009769525654576] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.11756483394447081, -0.05872123913100852] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8.E14	${\displaystyle{\displaystyle 2(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)=\frac{\omega^{2}-\alpha^{2}}{1+k^{\prime}}\Pi\left(\phi_{1},\alpha_{1}^{2},k_{1}\right)+k^{2}F\left(\phi,k\right)-{(1+k^{\prime})\alpha_{1}^{2}R_{C}\left(c_{1},c_{1}-\alpha_{1}^{2}\right)}}}$ 2(k^{2}-\alpha^{2})\incellintPik@{\phi}{\alpha^{2}}{k} = \frac{\omega^{2}-\alpha^{2}}{1+k^{\prime}}\incellintPik@{\phi_{1}}{\alpha_{1}^{2}}{k_{1}}+k^{2}\incellintFk@{\phi}{k}-{(1+k^{\prime})\alpha_{1}^{2}\CarlsonellintRC@{c_{1}}{c_{1}-\alpha_{1}^{2}}}		Error	2*((k)^(2)- \[Alpha]^(2))*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Divide[\[Omega]^(2)- \[Alpha]^(2),1 +Sqrt[1 - (k)^(2)]]*EllipticPi[(Subscript[\[Alpha], 1])^(2), \[Phi]+ ArcTan[Sqrt[1 - (k)^(2)]*Tan[\[Phi]]],(Subscript[k, 1])^2]+ (k)^(2)* EllipticF[\[Phi], (k)^2]-(1 +Sqrt[1 - (k)^(2)])*(Subscript[\[Alpha], 1])^(2)*1/Sqrt[((Csc[Subscript[\[Phi], 1]])^(2))- (Subscript[\[Alpha], 1])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((Csc[Subscript[\[Phi], 1]])^(2))/(((Csc[Subscript[\[Phi], 1]])^(2))- (Subscript[\[Alpha], 1])^(2))]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Complex[-1.4115811709537147, -1.2227387134851169] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[α, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.5976966939439394, -1.230515427208163] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ω, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[α, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex17	${\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{1+k}F\left(\phi_{2},k_{2}\right)}}$ \incellintFk@{\phi}{k} = \frac{2}{1+k}\incellintFk@{\phi_{2}}{k_{2}}		EllipticF(sin(phi), k) = (2)/(1 + k)*EllipticF(sin(phi[2]), k[2])	EllipticF[\[Phi], (k)^2] == Divide[2,1 + k]*EllipticF[Subscript[\[Phi], 2], (Subscript[k, 2])^2]	Failure	Failure	Failed [300 / 300] Result: .716161018e-1+.1278882161*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.163142760e-1+.3519262665*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.0030858847214221274, 0.01883659064247678] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.11075679050380455, 0.16335572999260056] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex18	${\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k)E\left(\phi_{2},k_{2}\right)+(1-k)F\left(\phi_{2},k_{2}\right)-k\sin\phi}}$ \incellintEk@{\phi}{k} = (1+k)\incellintEk@{\phi_{2}}{k_{2}}+(1-k)\incellintFk@{\phi_{2}}{k_{2}}-k\sin@@{\phi}		EllipticE(sin(phi), k) = (1 + k)*EllipticE(sin(phi[2]), k[2])+(1 - k)*EllipticF(sin(phi[2]), k[2])- k*sin(phi)	EllipticE[\[Phi], (k)^2] == (1 + k)*EllipticE[Subscript[\[Phi], 2], (Subscript[k, 2])^2]+(1 - k)*EllipticF[Subscript[\[Phi], 2], (Subscript[k, 2])^2]- k*Sin[\[Phi]]	Failure	Failure	Failed [300 / 300] Result: -.251128463-.1652679776*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 1} Result: .549972877-.903450862e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, phi[2] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.009026229866885283, -0.03603907810261833] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.42447097038130677, 0.1345883883024661] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ϕ, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex22	${\displaystyle{\displaystyle F\left(\phi,k\right)=(1+k_{1})F\left(\psi_{1},k_{1}\right)}}$ \incellintFk@{\phi}{k} = (1+k_{1})\incellintFk@{\psi_{1}}{k_{1}}		EllipticF(sin(phi), k) = (1 + k[1])*EllipticF(sin(psi[1]), k[1])	EllipticF[\[Phi], (k)^2] == (1 + Subscript[k, 1])*EllipticF[Subscript[\[Psi], 1], (Subscript[k, 1])^2]	Failure	Failure	Failed [299 / 300] Result: -.3025119160-.7226109033*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.6401936029-.6817361311*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [299 / 300] Result: Complex[-0.11940620612760577, -0.19771875715838422] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.15464125790413003, -0.13739720920586462] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8#Ex23	${\displaystyle{\displaystyle E\left(\phi,k\right)=(1+k^{\prime})E\left(\psi_{1},k_{1}\right)-k^{\prime}F\left(\phi,k\right)+(1-\Delta)\cot\phi}}$ \incellintEk@{\phi}{k} = (1+k^{\prime})\incellintEk@{\psi_{1}}{k_{1}}-k^{\prime}\incellintFk@{\phi}{k}+(1-\Delta)\cot@@{\phi}		EllipticE(sin(phi), k) = (1 +sqrt(1 - (k)^(2)))*EllipticE(sin(psi[1]), k[1])-sqrt(1 - (k)^(2))*EllipticF(sin(phi), k)+(1 -(sqrt(1 - (k)^(2)* (sin(phi))^(2))))*cot(phi)	EllipticE[\[Phi], (k)^2] == (1 +Sqrt[1 - (k)^(2)])*EllipticE[Subscript[\[Psi], 1], (Subscript[k, 1])^2]-Sqrt[1 - (k)^(2)]*EllipticF[\[Phi], (k)^2]+(1 -(Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]))*Cot[\[Phi]]	Failure	Failure	Failed [300 / 300] Result: -.5555013192-.1267358774*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 1} Result: -1.589246368-2.046785663*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k[1] = 1/2*3^(1/2)+1/2*I, psi[1] = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.22091089534718378, -0.1170454776590783] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.9299237807056446, -0.7272990802320405] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[k, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[ψ, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.8.E20	${\displaystyle{\displaystyle\rho\Pi\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k^{\prime}}\Pi\left(\psi_{1},\alpha_{1}^{2},k_{1}\right)+(\rho-1)F\left(\phi,k\right)-R_{C}\left(c-1,c-\alpha^{2}\right)}}$ \rho\incellintPik@{\phi}{\alpha^{2}}{k} = \frac{4}{1+k^{\prime}}\incellintPik@{\psi_{1}}{\alpha_{1}^{2}}{k_{1}}+(\rho-1)\incellintFk@{\phi}{k}-\CarlsonellintRC@{c-1}{c-\alpha^{2}}		Error	\[Rho]*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == Divide[4,1 +Sqrt[1 - (k)^(2)]]*EllipticPi[(Subscript[\[Alpha], 1])^(2), Subscript[\[Psi], 1],(Subscript[k, 1])^2]+(\[Rho]- 1)*EllipticF[\[Phi], (k)^2]- 1/Sqrt[((Csc[\[Phi]])^(2))- \[Alpha]^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(((Csc[\[Phi]])^(2))- 1)/(((Csc[\[Phi]])^(2))- \[Alpha]^(2))]	Missing Macro Error	Failure	-	Skipped - Because timed out
19.9#Ex1	${\displaystyle{\displaystyle\ln 4\leq K\left(k\right)+\ln k^{\prime}}}$ \ln@@{4} \leq \compellintKk@{k}+\ln@@{k^{\prime}}		ln(4) <= EllipticK(k)+ ln(sqrt(1 - (k)^(2)))	Log[4] <= EllipticK[(k)^2]+ Log[Sqrt[1 - (k)^(2)]]	Failure	Failure	Error	Failed [3 / 3] Result: LessEqual[1.3862943611198906, Indeterminate] Test Values: {Rule[k, 1]} Result: LessEqual[1.3862943611198906, Complex[1.392181321740353, 0.49253850304507485]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9#Ex1	${\displaystyle{\displaystyle K\left(k\right)+\ln k^{\prime}\leq\pi/2}}$ \compellintKk@{k}+\ln@@{k^{\prime}} \leq \pi/2		EllipticK(k)+ ln(sqrt(1 - (k)^(2))) <= Pi/2	EllipticK[(k)^2]+ Log[Sqrt[1 - (k)^(2)]] <= Pi/2	Failure	Failure	Error	Failed [3 / 3] Result: LessEqual[Indeterminate, 1.5707963267948966] Test Values: {Rule[k, 1]} Result: LessEqual[Complex[1.392181321740353, 0.49253850304507485], 1.5707963267948966] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9#Ex2	${\displaystyle{\displaystyle 1\leq E\left(k\right)}}$ 1 \leq \compellintEk@{k}		1 <= EllipticE(k)	1 <= EllipticE[(k)^2]	Failure	Failure	Successful [Tested: 3]	Failed [2 / 3] Result: LessEqual[1.0, Complex[0.40629888645996043, 1.343854231387098]] Test Values: {Rule[k, 2]} Result: LessEqual[1.0, Complex[0.2655964076372759, 2.498348127732516]] Test Values: {Rule[k, 3]}
19.9#Ex2	${\displaystyle{\displaystyle E\left(k\right)\leq\pi/2}}$ \compellintEk@{k} \leq \pi/2		EllipticE(k) <= Pi/2	EllipticE[(k)^2] <= Pi/2	Failure	Failure	Successful [Tested: 3]	Failed [2 / 3] Result: LessEqual[Complex[0.40629888645996043, 1.343854231387098], 1.5707963267948966] Test Values: {Rule[k, 2]} Result: LessEqual[Complex[0.2655964076372759, 2.498348127732516], 1.5707963267948966] Test Values: {Rule[k, 3]}
19.9#Ex3	${\displaystyle{\displaystyle 1\leq(2/\pi)\sqrt{1-\alpha^{2}}\Pi\left(\alpha^{2},k\right)\leq 1/k^{\prime}}}$ 1 \leq (2/\pi)\sqrt{1-\alpha^{2}}\compellintPik@{\alpha^{2}}{k}\leq 1/k^{\prime}	${\displaystyle{\displaystyle\alpha^{2}<1}}$	1 <= (2/Pi)*sqrt(1 - (alpha)^(2))*EllipticPi((alpha)^(2), k) <= 1/(sqrt(1 - (k)^(2)))	1 <= (2/Pi)*Sqrt[1 - \[Alpha]^(2)]*EllipticPi[\[Alpha]^(2), (k)^2] <= 1/(Sqrt[1 - (k)^(2)])	Failure	Failure	Error	Failed [3 / 3] Result: LessEqual[1.0, DirectedInfinity[], DirectedInfinity[]] Test Values: {Rule[k, 1], Rule[α, 0.5]} Result: LessEqual[1.0, Complex[0.4804983499812288, -0.6957733039705274], Complex[0.0, -0.5773502691896258]] Test Values: {Rule[k, 2], Rule[α, 0.5]} ... skip entries to safe data
19.9.E2	${\displaystyle{\displaystyle 1+\frac{{k^{\prime}}^{2}}{8}<\frac{K\left(k\right)}{\ln\left(4/k^{\prime}\right)}}}$ 1+\frac{{k^{\prime}}^{2}}{8} < \frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}}		1 +(1 - (k)^(2))/(8) < (EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2)))))	1 +Divide[1 - (k)^(2),8] < Divide[EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]]	Failure	Failure	Error	Failed [3 / 3] Result: Less[1.0, Indeterminate] Test Values: {Rule[k, 1]} Result: Less[0.625, Complex[0.7573351019929213, 0.13305010797062605]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E2	${\displaystyle{\displaystyle\frac{K\left(k\right)}{\ln\left(4/k^{\prime}\right)}<1+\frac{{k^{\prime}}^{2}}{4}}}$ \frac{\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 1+\frac{{k^{\prime}}^{2}}{4}		(EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2))))) < 1 +(1 - (k)^(2))/(4)	Divide[EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]] < 1 +Divide[1 - (k)^(2),4]	Failure	Failure	Error	Failed [3 / 3] Result: Less[Indeterminate, 1.0] Test Values: {Rule[k, 1]} Result: Less[Complex[0.7573351019929213, 0.13305010797062605], 0.25] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E3	${\displaystyle{\displaystyle 9+\frac{k^{2}{k^{\prime}}^{2}}{8}<\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k^{\prime}\right)}}}$ 9+\frac{k^{2}{k^{\prime}}^{2}}{8} < \frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}}		9 +((k)^(2)*1 - (k)^(2))/(8) < ((8 + (k)^(2))*EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2)))))	9 +Divide[(k)^(2)*1 - (k)^(2),8] < Divide[(8 + (k)^(2))*EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]]	Failure	Failure	Error	Failed [3 / 3] Result: Less[9.0, Indeterminate] Test Values: {Rule[k, 1]} Result: Less[9.0, Complex[9.088021223915057, 1.5966012956475137]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E3	${\displaystyle{\displaystyle\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k^{\prime}\right)}<9.096}}$ \frac{(8+k^{2})\compellintKk@{k}}{\ln@{4/k^{\prime}}} < 9.096		((8 + (k)^(2))*EllipticK(k))/(ln(4/(sqrt(1 - (k)^(2))))) < 9.096	Divide[(8 + (k)^(2))*EllipticK[(k)^2],Log[4/(Sqrt[1 - (k)^(2)])]] < 9.096	Failure	Failure	Error	Failed [3 / 3] Result: Less[Indeterminate, 9.096] Test Values: {Rule[k, 1]} Result: Less[Complex[9.088021223915057, 1.5966012956475137], 9.096] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E4	${\displaystyle{\displaystyle\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3}\leq\frac{2}{\pi}E\left(k\right)}}$ \left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3} \leq \frac{2}{\pi}\compellintEk@{k}		((1 +(sqrt(1 - (k)^(2)))^(3/2))/(2))^(2/3) <= (2)/(Pi)*EllipticE(k)	(Divide[1 +(Sqrt[1 - (k)^(2)])^(3/2),2])^(2/3) <= Divide[2,Pi]*EllipticE[(k)^2]	Failure	Failure	Successful [Tested: 3]	Failed [2 / 3] Result: LessEqual[Complex[0.2518251425072316, 0.8700591952646104], Complex[0.2586579046113418, 0.8555241748808654]] Test Values: {Rule[k, 2]} Result: LessEqual[Complex[0.1858923839966674, 1.6059081831429025], Complex[0.16908392457168991, 1.5904978163720476]] Test Values: {Rule[k, 3]}
19.9.E4	${\displaystyle{\displaystyle\frac{2}{\pi}E\left(k\right)\leq\left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}}}$ \frac{2}{\pi}\compellintEk@{k} \leq \left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}		(2)/(Pi)*EllipticE(k) <= ((1 +1 - (k)^(2))/(2))^(1/2)	Divide[2,Pi]*EllipticE[(k)^2] <= (Divide[1 +1 - (k)^(2),2])^(1/2)	Failure	Failure	Successful [Tested: 3]	Failed [2 / 3] Result: LessEqual[Complex[0.2586579046113418, 0.8555241748808654], Complex[0.0, 1.0]] Test Values: {Rule[k, 2]} Result: LessEqual[Complex[0.16908392457168991, 1.5904978163720476], Complex[0.0, 1.8708286933869707]] Test Values: {Rule[k, 3]}
19.9.E5	${\displaystyle{\displaystyle\ln\frac{(1+\sqrt{k^{\prime}})^{2}}{k}<\frac{\pi{K^{\prime}}\left(k\right)}{2K\left(k\right)}}}$ \ln@@{\frac{(1+\sqrt{k^{\prime}})^{2}}{k}} < \frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}}		ln(((1 +sqrt(sqrt(1 - (k)^(2))))^(2))/(k)) < (Pi*EllipticCK(k))/(2*EllipticK(k))	Log[Divide[(1 +Sqrt[Sqrt[1 - (k)^(2)]])^(2),k]] < Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]]	Failure	Failure	Error	Failed [3 / 3] Result: False Test Values: {Rule[k, 1]} Result: Less[Complex[0.8314429455293103, 0.8983332083070389], Complex[0.762166367418117, 0.9750101446769989]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E5	${\displaystyle{\displaystyle\frac{\pi{K^{\prime}}\left(k\right)}{2K\left(k\right)}<\ln\frac{2(1+k^{\prime})}{k}}}$ \frac{\pi\ccompellintKk@{k}}{2\compellintKk@{k}} < \ln@@{\frac{2(1+k^{\prime})}{k}}		(Pi*EllipticCK(k))/(2*EllipticK(k)) < ln((2*(1 +sqrt(1 - (k)^(2))))/(k))	Divide[Pi*EllipticK[1-(k)^2],2*EllipticK[(k)^2]] < Log[Divide[2*(1 +Sqrt[1 - (k)^(2)]),k]]	Failure	Failure	Error	Failed [2 / 3] Result: Less[Complex[0.762166367418117, 0.9750101446769989], Complex[0.6931471805599452, 1.0471975511965976]] Test Values: {Rule[k, 2]} Result: Less[Complex[0.7130154358988758, 1.1147297033963086], Complex[0.6931471805599453, 1.2309594173407747]] Test Values: {Rule[k, 3]}
19.9.E6	${\displaystyle{\displaystyle(1-\tfrac{3}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k\right))}}$ (1-\tfrac{3}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k})		(1 -(3)/(4)*(k)^(2))^(- 1/2) < (4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k))	(1 -Divide[3,4]*(k)^(2))^(- 1/2) < Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Less[2.0, DirectedInfinity[]] Test Values: {Rule[k, 1]} Result: Less[Complex[0.0, -0.7071067811865475], Complex[0.13896654948167025, -0.7709822125950203]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E6	${\displaystyle{\displaystyle\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k\right))<(k^{\prime})^{-3/4}}}$ \frac{4}{\pi k^{2}}(\compellintKk@{k}-\compellintEk@{k}) < (k^{\prime})^{-3/4}		(4)/(Pi*(k)^(2))*(EllipticK(k)- EllipticE(k)) < (sqrt(1 - (k)^(2)))^(- 3/4)	Divide[4,Pi*(k)^(2)]*(EllipticK[(k)^2]- EllipticE[(k)^2]) < (Sqrt[1 - (k)^(2)])^(- 3/4)	Failure	Failure	Error	Failed [3 / 3] Result: Less[DirectedInfinity[], DirectedInfinity[]] Test Values: {Rule[k, 1]} Result: Less[Complex[0.13896654948167025, -0.7709822125950203], Complex[0.2534656958546175, -0.6119203205285516]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E7	${\displaystyle{\displaystyle(1-\tfrac{1}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}}$ (1-\tfrac{1}{4}k^{2})^{-1/2} < \frac{4}{\pi k^{2}}(\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k})		(1 -(1)/(4)*(k)^(2))^(- 1/2) < (4)/(Pi*(k)^(2))*(EllipticE(k)-1 - (k)^(2)*EllipticK(k))	(1 -Divide[1,4]*(k)^(2))^(- 1/2) < Divide[4,Pi*(k)^(2)]*(EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Less[1.1547005383792517, DirectedInfinity[]] Test Values: {Rule[k, 1]} Result: Less[DirectedInfinity[], Complex[-1.2621629410274844, 1.800642588058783]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E8	${\displaystyle{\displaystyle k^{\prime}<\frac{E\left(k\right)}{K\left(k\right)}}}$ k^{\prime} < \frac{\compellintEk@{k}}{\compellintKk@{k}}		sqrt(1 - (k)^(2)) < (EllipticE(k))/(EllipticK(k))	Sqrt[1 - (k)^(2)] < Divide[EllipticE[(k)^2],EllipticK[(k)^2]]	Failure	Failure	Error	Failed [3 / 3] Result: False Test Values: {Rule[k, 1]} Result: Less[Complex[0.0, 1.7320508075688772], Complex[-0.5907718728609501, 0.8386174564999851]] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.9.E8	${\displaystyle{\displaystyle\frac{E\left(k\right)}{K\left(k\right)}<\left(\frac{1+k^{\prime}}{2}\right)^{2}}}$ \frac{\compellintEk@{k}}{\compellintKk@{k}} < \left(\frac{1+k^{\prime}}{2}\right)^{2}		(EllipticE(k))/(EllipticK(k)) < ((1 +sqrt(1 - (k)^(2)))/(2))^(2)	Divide[EllipticE[(k)^2],EllipticK[(k)^2]] < (Divide[1 +Sqrt[1 - (k)^(2)],2])^(2)	Failure	Failure	Error	Failed [2 / 3] Result: Less[Complex[-0.5907718728609501, 0.8386174564999851], Complex[-0.4999999999999999, 0.8660254037844386]] Test Values: {Rule[k, 2]} Result: Less[Complex[-1.9604512687154212, 1.5690726247192568], Complex[-1.7500000000000004, 1.4142135623730951]] Test Values: {Rule[k, 3]}
19.9.E9	${\displaystyle{\displaystyle L(a,b)=4aE\left(k\right)}}$ L(a,b) = 4a\compellintEk@{k}	${\displaystyle{\displaystyle k^{2}=1-(b^{2}/a^{2}),a>b}}$	L(a , b) = 4*a*EllipticE(k)	L[a , b] == 4*a*EllipticE[(k)^2]	Error	Failure	-	Error
19.9.E11	${\displaystyle{\displaystyle\phi\leq F\left(\phi,k\right)}}$ \phi \leq \incellintFk@{\phi}{k}		phi <= EllipticF(sin(phi), k)	\[Phi] <= EllipticF[\[Phi], (k)^2]	Failure	Failure	Failed [4 / 30] Result: -1.500000000 <= -3.340677542 Test Values: {phi = -3/2, k = 1} Result: -.5000000000 <= -.5222381033 Test Values: {phi = -1/2, k = 1} ... skip entries to safe data	Failed [28 / 30] Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43180375739814203, 0.27142936483528934]] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.3965687056216178, 0.33175091278780894]] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E12	${\displaystyle{\displaystyle E\left(\phi,k\right)\leq\phi}}$ \incellintEk@{\phi}{k} \leq \phi		EllipticE(sin(phi), k) <= phi	EllipticE[\[Phi], (k)^2] <= \[Phi]	Failure	Failure	Failed [4 / 30] Result: -.9974949866 <= -1.500000000 Test Values: {phi = -3/2, k = 1} Result: -.4794255386 <= -.5000000000 Test Values: {phi = -1/2, k = 1} ... skip entries to safe data	Failed [27 / 30] Result: LessEqual[Complex[0.43278851685803155, 0.22929764467344024], Complex[0.43301270189221935, 0.24999999999999997]] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: LessEqual[Complex[0.44208095936294645, 0.16535187593702125], Complex[0.43301270189221935, 0.24999999999999997]] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E13	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},0\right)\leq\Pi\left(\phi,\alpha^{2},k\right)}}$ \incellintPik@{\phi}{\alpha^{2}}{0} \leq \incellintPik@{\phi}{\alpha^{2}}{k}		EllipticPi(sin(phi), (alpha)^(2), 0) <= EllipticPi(sin(phi), (alpha)^(2), k)	EllipticPi[\[Alpha]^(2), \[Phi],(0)^2] <= EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]	Failure	Failure	Failed [8 / 90] Result: -.6351972518 <= -.6692391842 Test Values: {alpha = 3/2, phi = -1/2, k = 1} Result: -.6351972518 <= -.9273807742 Test Values: {alpha = 3/2, phi = -1/2, k = 2} ... skip entries to safe data	Failed [84 / 90] Result: LessEqual[Complex[0.4032669574270382, 0.3492210121777662], Complex[0.39392267303966433, 0.37152709024037445]] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: LessEqual[Complex[0.4032669574270382, 0.3492210121777662], Complex[0.33490711362096304, 0.4200642464932446]] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E14	${\displaystyle{\displaystyle\frac{3}{1+\Delta+\cos\phi}<\frac{F\left(\phi,k\right)}{\sin\phi}}}$ \frac{3}{1+\Delta+\cos@@{\phi}} < \frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}}		(3)/(1 + Delta + cos(phi)) < (EllipticF(sin(phi), k))/(sin(phi))	Divide[3,1 + \[CapitalDelta]+ Cos[\[Phi]]] < Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]]	Failure	Failure	Failed [16 / 300] Result: 7.945282179 < 1.089299717 Test Values: {Delta = -3/2, phi = -1/2, k = 1} Result: 7.945282179 < 1.412977582 Test Values: {Delta = -3/2, phi = -1/2, k = 2} ... skip entries to safe data	Failed [284 / 300] Result: Less[Complex[1.261572446843062, -0.07667841479591199], Complex[1.0384958486950706, 0.07695378095553612]] Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Less[Complex[1.261572446843062, -0.07667841479591199], Complex[1.0325857379409573, 0.21946385233164167]] Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E14	${\displaystyle{\displaystyle\frac{F\left(\phi,k\right)}{\sin\phi}<\frac{1}{(\Delta\cos\phi)^{1/3}}}}$ \frac{\incellintFk@{\phi}{k}}{\sin@@{\phi}} < \frac{1}{(\Delta\cos@@{\phi})^{1/3}}		(EllipticF(sin(phi), k))/(sin(phi)) < (1)/((Delta*cos(phi))^(1/3))	Divide[EllipticF[\[Phi], (k)^2],Sin[\[Phi]]] < Divide[1,(\[CapitalDelta]*Cos[\[Phi]])^(1/3)]	Failure	Failure	Failed [20 / 300] Result: 1.675417084 < 1.170093898 Test Values: {Delta = -3/2, phi = -2, k = 1} Result: 1.675417084 < 1.170093898 Test Values: {Delta = -3/2, phi = 2, k = 1} ... skip entries to safe data	Failed [298 / 300] Result: Less[Complex[1.0384958486950706, 0.07695378095553612], Complex[1.2731409874856745, -0.17545913345292982]] Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Less[Complex[1.0325857379409573, 0.21946385233164167], Complex[1.2731409874856745, -0.17545913345292982]] Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E15	${\displaystyle{\displaystyle 1<F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos\phi}\right)\right)}}$ 1 < \incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right)		1 < EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi))))	1 < EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]])	Failure	Failure	Failed [6 / 300] Result: 1. < .4615167558 Test Values: {Delta = -1/2, phi = -1/2, k = 1} Result: 1. < .5986532627 Test Values: {Delta = -1/2, phi = -1/2, k = 2} ... skip entries to safe data	Failed [288 / 300] Result: Less[1.0, Complex[0.9573719244599448, 0.16621131448588694]] Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Less[1.0, Complex[0.9388814261604885, 0.2980132161872323]] Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E15	${\displaystyle{\displaystyle F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos\phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin^{2}}\phi}}}$ \incellintFk@{\phi}{k}\bigg{/}\left((\sin@@{\phi})\ln@{\frac{4}{\Delta+\cos@@{\phi}}}\right) < \frac{4}{2+(1+k^{2})\sin^{2}@@{\phi}}		EllipticF(sin(phi), k)/((sin(phi))*ln((4)/(Delta + cos(phi)))) < (4)/(2 +(1 + (k)^(2))*(sin(phi))^(2))	EllipticF[\[Phi], (k)^2]/((Sin[\[Phi]])*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]]) < Divide[4,2 +(1 + (k)^(2))*(Sin[\[Phi]])^(2)]	Failure	Failure	Failed [20 / 300] Result: 3.582850518 < 1.002508151 Test Values: {Delta = 3/2, phi = -3/2, k = 1} Result: 3.582850518 < 1.002508151 Test Values: {Delta = 3/2, phi = 3/2, k = 1} ... skip entries to safe data	Failed [296 / 300] Result: Less[Complex[0.9573719244599448, 0.16621131448588694], Complex[1.7102149955099495, -0.29913282294542826]] Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Less[Complex[0.9388814261604885, 0.2980132161872323], Complex[1.3149325512421652, -0.4880625346303866]] Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E16	${\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{\pi}K\left(k^{\prime}\right)\ln\left(\frac{4}{\Delta+\cos\phi}\right)-\theta\Delta^{2}}}$ \incellintFk@{\phi}{k} = \frac{2}{\pi}\compellintKk@{k^{\prime}}\ln@{\frac{4}{\Delta+\cos@@{\phi}}}-\theta\Delta^{2}	${\displaystyle{\displaystyle(\sin\phi)/8<\theta,\theta<(\ln 2)/(k^{2}\sin\phi)}}$	EllipticF(sin(phi), k) = (2)/(Pi)*EllipticK(sqrt(1 - (k)^(2)))*ln((4)/(Delta + cos(phi)))- theta*(Delta)^(2)	EllipticF[\[Phi], (k)^2] == Divide[2,Pi]*EllipticK[(Sqrt[1 - (k)^(2)])^2]*Log[Divide[4,\[CapitalDelta]+ Cos[\[Phi]]]]- \[Theta]*\[CapitalDelta]^(2)	Failure	Failure	Failed [30 / 30] Result: 2.264395299+.9232968251*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 3/2, theta = 1/2, k = 1} Result: -.185868314e-1+.7122804653*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2, theta = 1/2, k = 1} ... skip entries to safe data	Failed [30 / 30] Result: Complex[1.4412941413043292, 0.5689187621917111] Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 0.5], Rule[ϕ, 1.5]} Result: Complex[-0.5132046492108906, 0.2967418012807382] Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, 0.5], Rule[ϕ, 0.5]} ... skip entries to safe data
19.9.E17	${\displaystyle{\displaystyle L\leq F\left(\phi,k\right)}}$ L \leq \incellintFk@{\phi}{k}		L <= EllipticF(sin(phi), k)	L <= EllipticF[\[Phi], (k)^2]	Failure	Failure	Failed [24 / 300] Result: -1.500000000 <= -3.340677542 Test Values: {L = -3/2, phi = -3/2, k = 1} Result: -1.500000000 <= -1.523452443 Test Values: {L = -3/2, phi = -2, k = 1} ... skip entries to safe data	Failed [288 / 300] Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43180375739814203, 0.27142936483528934]] Test Values: {Rule[k, 1], Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.3965687056216178, 0.33175091278780894]] Test Values: {Rule[k, 2], Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E17	${\displaystyle{\displaystyle F\left(\phi,k\right)\leq\sqrt{UL}}}$ \incellintFk@{\phi}{k} \leq \sqrt{UL}		EllipticF(sin(phi), k) <= sqrt(U*L)	EllipticF[\[Phi], (k)^2] <= Sqrt[U*L]	Failure	Failure	Successful [Tested: 300]	Failed [300 / 300] Result: LessEqual[Complex[0.43180375739814203, 0.27142936483528934], Complex[0.43301270189221935, 0.24999999999999997]] Test Values: {Rule[k, 1], Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: LessEqual[Complex[0.3965687056216178, 0.33175091278780894], Complex[0.43301270189221935, 0.24999999999999997]] Test Values: {Rule[k, 2], Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.9.E17	${\displaystyle{\displaystyle\sqrt{UL}\leq\tfrac{1}{2}(U+L)}}$ \sqrt{UL} \leq \tfrac{1}{2}(U+L)		sqrt(U*L) <= (1)/(2)*(U + L)	Sqrt[U*L] <= Divide[1,2]*(U + L)	Failure	Failure	Failed [9 / 100] Result: 1.500000000 <= -1.500000000 Test Values: {L = -3/2, U = -3/2} Result: .8660254040 <= -1. Test Values: {L = -3/2, U = -1/2} ... skip entries to safe data	Failed [91 / 100] Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43301270189221935, 0.24999999999999997]] Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: LessEqual[Complex[0.12940952255126037, 0.48296291314453416], Complex[0.09150635094610973, 0.34150635094610965]] Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.9.E17	${\displaystyle{\displaystyle\tfrac{1}{2}(U+L)\leq U}}$ \tfrac{1}{2}(U+L) \leq U		(1)/(2)*(U + L) <= U	Divide[1,2]*(U + L) <= U	Failure	Failure	Failed [15 / 100] Result: -1.750000000 <= -2. Test Values: {L = -3/2, U = -2} Result: 0. <= -1.500000000 Test Values: {L = 3/2, U = -3/2} ... skip entries to safe data	Failed [79 / 100] Result: LessEqual[Complex[0.43301270189221935, 0.24999999999999997], Complex[0.43301270189221935, 0.24999999999999997]] Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: LessEqual[Complex[0.09150635094610973, 0.34150635094610965], Complex[-0.2499999999999999, 0.43301270189221935]] Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.9#Ex4	${\displaystyle{\displaystyle L=(1/\sigma)\operatorname{arctanh}\left(\sigma\sin\phi\right)}}$ L = (1/\sigma)\atanh@{\sigma\sin@@{\phi}}	${\displaystyle{\displaystyle\sigma=\sqrt{(1+k^{2})/2}}}$	L = (1/sigma)*arctanh(sigma*sin(phi))	L == (1/\[Sigma])*ArcTanh[\[Sigma]*Sin[\[Phi]]]	Failure	Failure	Failed [300 / 300] Result: .1841715885+.458206673e-1*I Test Values: {L = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I} Result: -.197696883e-1+.4084290873*I Test Values: {L = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, sigma = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.008169183554908921, 0.015254361571334585] Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[σ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.6990489693230986, -0.19299436497537428] Test Values: {Rule[L, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[σ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.9#Ex5	${\displaystyle{\displaystyle U=\tfrac{1}{2}\operatorname{arctanh}\left(\sin\phi\right)+\tfrac{1}{2}k^{-1}\operatorname{arctanh}\left(k\sin\phi\right)}}$ U = \tfrac{1}{2}\atanh@{\sin@@{\phi}}+\tfrac{1}{2}k^{-1}\atanh@{k\sin@@{\phi}}		U = (1)/(2)*arctanh(sin(phi))+(1)/(2)*(k)^(- 1)* arctanh(k*sin(phi))	U == Divide[1,2]*ArcTanh[Sin[\[Phi]]]+Divide[1,2]*(k)^(- 1)* ArcTanh[k*Sin[\[Phi]]]	Failure	Failure	Failed [300 / 300] Result: .451553750e-1-.1773780507*I Test Values: {U = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 1} Result: .3250459090-.1674857034*I Test Values: {U = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.0012089444940770466, -0.021429364835289427] Test Values: {Rule[k, 1], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.04320077983427789, -0.07655275524887523] Test Values: {Rule[k, 2], Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.10#Ex1	${\displaystyle{\displaystyle\ln\left(x/y\right)=(x-y)R_{C}\left(\tfrac{1}{4}(x+y)^{2},xy\right)}}$ \ln@{x/y} = (x-y)\CarlsonellintRC@{\tfrac{1}{4}(x+y)^{2}}{xy}		Error	Log[x/y] == (x - y)*1/Sqrt[x*y]*Hypergeometric2F1[1/2,1/2,3/2,1-(Divide[1,4]*(x + y)^(2))/(x*y)]	Missing Macro Error	Failure	-	Failed [12 / 18] Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10#Ex2	${\displaystyle{\displaystyle\operatorname{arctan}\left(x/y\right)=xR_{C}\left(y^{2},y^{2}+x^{2}\right)}}$ \atan@{x/y} = x\CarlsonellintRC@{y^{2}}{y^{2}+x^{2}}		Error	ArcTan[x/y] == x*1/Sqrt[(y)^(2)+ (x)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2))/((y)^(2)+ (x)^(2))]	Missing Macro Error	Failure	-	Failed [9 / 18] Result: -1.5707963267948966 Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: -2.498091544796509 Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.10#Ex3	${\displaystyle{\displaystyle\operatorname{arctanh}\left(x/y\right)=xR_{C}\left(y^{2},y^{2}-x^{2}\right)}}$ \atanh@{x/y} = x\CarlsonellintRC@{y^{2}}{y^{2}-x^{2}}		Error	ArcTanh[x/y] == x*1/Sqrt[(y)^(2)- (x)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2))/((y)^(2)- (x)^(2))]	Missing Macro Error	Failure	-	Failed [15 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10#Ex4	${\displaystyle{\displaystyle\operatorname{arcsin}\left(x/y\right)=xR_{C}\left(y^{2}-x^{2},y^{2}\right)}}$ \asin@{x/y} = x\CarlsonellintRC@{y^{2}-x^{2}}{y^{2}}		Error	ArcSin[x/y] == x*1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2)- (x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [9 / 18] Result: -3.141592653589793 Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-3.141592653589793, 3.525494348078172] Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.10#Ex5	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(x/y\right)=xR_{C}\left(y^{2}+x^{2},y^{2}\right)}}$ \asinh@{x/y} = x\CarlsonellintRC@{y^{2}+x^{2}}{y^{2}}		Error	ArcSinh[x/y] == x*1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((y)^(2)+ (x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [9 / 18] Result: Complex[-1.7627471740390859, 0.0] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-3.6368929184641337, 0.0] Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.10#Ex6	${\displaystyle{\displaystyle\operatorname{arccos}\left(x/y\right)=(y^{2}-x^{2})^{1/2}R_{C}\left(x^{2},y^{2}\right)}}$ \acos@{x/y} = (y^{2}-x^{2})^{1/2}\CarlsonellintRC@{x^{2}}{y^{2}}		Error	ArcCos[x/y] == ((y)^(2)- (x)^(2))^(1/2)* 1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [12 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10#Ex7	${\displaystyle{\displaystyle\operatorname{arccosh}\left(x/y\right)=(x^{2}-y^{2})^{1/2}R_{C}\left(x^{2},y^{2}\right)}}$ \acosh@{x/y} = (x^{2}-y^{2})^{1/2}\CarlsonellintRC@{x^{2}}{y^{2}}		Error	ArcCosh[x/y] == ((x)^(2)- (y)^(2))^(1/2)* 1/Sqrt[(y)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-((x)^(2))/((y)^(2))]	Missing Macro Error	Failure	-	Failed [12 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.10.E2	${\displaystyle{\displaystyle(\sinh\phi)R_{C}\left(1,{\cosh^{2}}\phi\right)=\operatorname{gd}\left(\phi\right)}}$ (\sinh@@{\phi})\CarlsonellintRC@{1}{\cosh^{2}@@{\phi}} = \Gudermannian@{\phi}	${\displaystyle{\displaystyle-\infty<(\phi),(\phi)<\infty}}$	Error	(Sinh[\[Phi]])*1/Sqrt[(Cosh[\[Phi]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(1)/((Cosh[\[Phi]])^(2))] == Gudermannian[\[Phi]]	Missing Macro Error	Failure	-	Successful [Tested: 6]
19.11.E1	${\displaystyle{\displaystyle F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right)}}$ \incellintFk@{\theta}{k}+\incellintFk@{\phi}{k} = \incellintFk@{\psi}{k}		EllipticF(sin(theta), k)+ EllipticF(sin(phi), k) = EllipticF(sin(psi), k)	EllipticF[\[Theta], (k)^2]+ EllipticF[\[Phi], (k)^2] == EllipticF[\[Psi], (k)^2]	Failure	Failure	Failed [300 / 300] Result: .8208700290+.6773780507*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: .4831883421+.7182528229*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.43180375739814203, 0.27142936483528934] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.3965687056216178, 0.33175091278780894] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11.E2	${\displaystyle{\displaystyle E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin\theta\sin\phi\sin\psi}}$ \incellintEk@{\theta}{k}+\incellintEk@{\phi}{k} = \incellintEk@{\psi}{k}+k^{2}\sin@@{\theta}\sin@@{\phi}\sin@@{\psi}		EllipticE(sin(theta), k)+ EllipticE(sin(phi), k) = EllipticE(sin(psi), k)+ (k)^(2)* sin(theta)*sin(phi)*sin(psi)	EllipticE[\[Theta], (k)^2]+ EllipticE[\[Phi], (k)^2] == EllipticE[\[Psi], (k)^2]+ (k)^(2)* Sin[\[Theta]]*Sin[\[Phi]]*Sin[\[Psi]]	Failure	Failure	Failed [300 / 300] Result: .5188815884-.3712110352*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.324003006-2.889566484*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.41998937174924766, 0.11250711558240023] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.3908843789278109, -0.3018102404271388] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11#Ex3	${\displaystyle{\displaystyle\cos\psi=\frac{\cos\theta\cos\phi-(\sin\theta\sin\phi)\Delta(\theta)\Delta(\phi)}{1-k^{2}{\sin^{2}}\theta{\sin^{2}}\phi}}}$ \cos@@{\psi} = \frac{\cos@@{\theta}\cos@@{\phi}-(\sin@@{\theta}\sin@@{\phi})\Delta(\theta)\Delta(\phi)}{1-k^{2}\sin^{2}@@{\theta}\sin^{2}@@{\phi}}		cos(psi) = (cos(theta)*cos(phi)-(sin(theta)*sin(phi))*(sqrt(1 - (k)^(2)* (sin(theta))^(2)))*Delta(phi))/(1 - (k)^(2)* (sin(theta))^(2)* (sin(phi))^(2))	Cos[\[Psi]] == Divide[Cos[\[Theta]]*Cos[\[Phi]]-(Sin[\[Theta]]*Sin[\[Phi]])*(Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)])*\[CapitalDelta][\[Phi]],1 - (k)^(2)* (Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2)]	Failure	Failure	Failed [300 / 300] Result: -.360132946e-1+.3498736067*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: .3023079579-.441042741e-1*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.06008432780660544, 0.09466439987688165] Test Values: {Rule[k, 1], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.1274461431695849, -0.029704144406044533] Test Values: {Rule[k, 2], Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11#Ex4	${\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=\frac{(\sin\theta)\Delta(\phi)+(\sin\phi)\Delta(\theta)}{\cos\theta+\cos\phi}}}$ \tan@{\tfrac{1}{2}\psi} = \frac{(\sin@@{\theta})\Delta(\phi)+(\sin@@{\phi})\Delta(\theta)}{\cos@@{\theta}+\cos@@{\phi}}		tan((1)/(2)*psi) = ((sin(theta))*Delta(phi)+(sin(phi))*(sqrt(1 - (k)^(2)* (sin(theta))^(2))))/(cos(theta)+ cos(phi))	Tan[Divide[1,2]*\[Psi]] == Divide[(Sin[\[Theta]])*\[CapitalDelta][\[Phi]]+(Sin[\[Phi]])*(Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)]),Cos[\[Theta]]+ Cos[\[Phi]]]	Translation Error	Translation Error	-	-
19.11.E5	${\displaystyle{\displaystyle\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right)}}$ \incellintPik@{\theta}{\alpha^{2}}{k}+\incellintPik@{\phi}{\alpha^{2}}{k} = \incellintPik@{\psi}{\alpha^{2}}{k}-\alpha^{2}\CarlsonellintRC@{\gamma-\delta}{\gamma}		Error	EllipticPi[\[Alpha]^(2), \[Theta],(k)^2]+ EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == EllipticPi[\[Alpha]^(2), \[Psi],(k)^2]- \[Alpha]^(2)* 1/Sqrt[\[Gamma]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Gamma]-(\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))))/(\[Gamma])]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[2.431737700775111, 0.07689658395417326] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.648685299290325, -1.4197583822626343] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11.E6_5	${\displaystyle{\displaystyle R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-\alpha^{2}\cos\theta\cos\phi\cos\psi}\right)}}$ \CarlsonellintRC@{\gamma-\delta}{\gamma} = \frac{-1}{\sqrt{\delta}}\atan@{\frac{\sqrt{\delta}\sin@@{\theta}\sin@@{\phi}\sin@@{\psi}}{\alpha^{2}-1-\alpha^{2}\cos@@{\theta}\cos@@{\phi}\cos@@{\psi}}}		Error	1/Sqrt[\[Gamma]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Gamma]-(\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))))/(\[Gamma])] == Divide[- 1,Sqrt[\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))]]*ArcTan[Divide[Sqrt[\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))]*Sin[\[Theta]]*Sin[\[Phi]]*Sin[\[Psi]],\[Alpha]^(2)- 1 - \[Alpha]^(2)* Cos[\[Theta]]*Cos[\[Phi]]*Cos[\[Psi]]]]	Missing Macro Error	Translation Error	-	-
19.11.E7	${\displaystyle{\displaystyle F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right)}}$ \incellintFk@{\phi}{k} = \compellintKk@{k}-\incellintFk@{\theta}{k}		EllipticF(sin(phi), k) = EllipticK(k)- EllipticF(sin(theta), k)	EllipticF[\[Phi], (k)^2] == EllipticK[(k)^2]- EllipticF[\[Theta], (k)^2]	Failure	Failure	Error	Failed [300 / 300] Result: DirectedInfinity[] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.04973776616306258, 1.7417596493254397] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11.E8	${\displaystyle{\displaystyle E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta\sin\phi}}$ \incellintEk@{\phi}{k} = \compellintEk@{k}-\incellintEk@{\theta}{k}+k^{2}\sin@@{\theta}\sin@@{\phi}		EllipticE(sin(phi), k) = EllipticE(k)- EllipticE(sin(theta), k)+ (k)^(2)* sin(theta)*sin(phi)	EllipticE[\[Phi], (k)^2] == EllipticE[(k)^2]- EllipticE[\[Theta], (k)^2]+ (k)^(2)* Sin[\[Theta]]*Sin[\[Phi]]	Failure	Failure	Failed [295 / 300] Result: .940848258e-1+.952154806e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.829018303-3.772436995*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [297 / 300] Result: Complex[-0.2691514567553243, 0.26012051423236426] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.06105092961961717, -1.8070495799711206] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11.E9	${\displaystyle{\displaystyle\tan\theta=1/(k^{\prime}\tan\phi)}}$ \tan@@{\theta} = 1/(k^{\prime}\tan@@{\phi})		tan(theta) = 1/(sqrt(1 - (k)^(2))*tan(phi))	Tan[\[Theta]] == 1/(Sqrt[1 - (k)^(2)]*Tan[\[Phi]])	Failure	Failure	Failed [300 / 300] Result: Float(infinity)+Float(infinity)*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: 1.112198033+1.184536461*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: DirectedInfinity[] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.0561283793604441, 1.210195136063891] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11.E10	${\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)-\Pi\left(\theta,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right)}}$ \incellintPik@{\phi}{\alpha^{2}}{k} = \compellintPik@{\alpha^{2}}{k}-\incellintPik@{\theta}{\alpha^{2}}{k}-\alpha^{2}\CarlsonellintRC@{\gamma-\delta}{\gamma}		Error	EllipticPi[\[Alpha]^(2), \[Phi],(k)^2] == EllipticPi[\[Alpha]^(2), (k)^2]- EllipticPi[\[Alpha]^(2), \[Theta],(k)^2]- \[Alpha]^(2)* 1/Sqrt[\[Gamma]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Gamma]-(\[Alpha]^(2)*(1 - \[Alpha]^(2))*(\[Alpha]^(2)- (k)^(2))))/(\[Gamma])]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: DirectedInfinity[] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[2.2835000786563655, -0.476202278380103] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11.E12	${\displaystyle{\displaystyle F\left(\psi,k\right)=2F\left(\theta,k\right)}}$ \incellintFk@{\psi}{k} = 2\incellintFk@{\theta}{k}		EllipticF(sin(psi), k) = 2*EllipticF(sin(theta), k)	EllipticF[\[Psi], (k)^2] == 2*EllipticF[\[Theta], (k)^2]	Failure	Failure	Failed [300 / 300] Result: -.8208700290-.6773780507*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.4831883421-.7182528229*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.43180375739814203, -0.27142936483528934] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.3965687056216178, -0.33175091278780894] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11.E13	${\displaystyle{\displaystyle E\left(\psi,k\right)=2E\left(\theta,k\right)-k^{2}{\sin^{2}}\theta\sin\psi}}$ \incellintEk@{\psi}{k} = 2\incellintEk@{\theta}{k}-k^{2}\sin^{2}@@{\theta}\sin@@{\psi}		EllipticE(sin(psi), k) = 2*EllipticE(sin(theta), k)- (k)^(2)* (sin(theta))^(2)* sin(psi)	EllipticE[\[Psi], (k)^2] == 2*EllipticE[\[Theta], (k)^2]- (k)^(2)* (Sin[\[Theta]])^(2)* Sin[\[Psi]]	Failure	Failure	Failed [300 / 300] Result: -.5188815884+.3712110352*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: .324003006+2.889566484*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [298 / 300] Result: Complex[-0.41998937174924766, -0.11250711558240023] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.3908843789278109, 0.3018102404271388] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11#Ex9	${\displaystyle{\displaystyle\cos\psi=(\cos\left(2\theta\right)+k^{2}{\sin^{4}}\theta)/(1-k^{2}{\sin^{4}}\theta)}}$ \cos@@{\psi} = (\cos@{2\theta}+k^{2}\sin^{4}@@{\theta})/(1-k^{2}\sin^{4}@@{\theta})		cos(psi) = (cos(2*theta)+ (k)^(2)* (sin(theta))^(4))/(1 - (k)^(2)* (sin(theta))^(4))	Cos[\[Psi]] == (Cos[2*\[Theta]]+ (k)^(2)* (Sin[\[Theta]])^(4))/(1 - (k)^(2)* (Sin[\[Theta]])^(4))	Failure	Failure	Failed [300 / 300] Result: .6382547213-.68319321e-2*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: 1.291602175-.5372399851*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.22600457397095797, 0.19313483829287414] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.33144266284556045, -0.05654646036238595] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11#Ex10	${\displaystyle{\displaystyle\tan\left(\tfrac{1}{2}\psi\right)=(\tan\theta)\Delta(\theta)}}$ \tan@{\tfrac{1}{2}\psi} = (\tan@@{\theta})\Delta(\theta)		tan((1)/(2)*psi) = (tan(theta))*(sqrt(1 - (k)^(2)* (sin(theta))^(2)))	Tan[Divide[1,2]*\[Psi]] == (Tan[\[Theta]])*(Sqrt[1 - (k)^(2)* (Sin[\[Theta]])^(2)])	Failure	Failure	Failed [300 / 300] Result: -.4299370879-.441018886e-1*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 1} Result: -1.378631246+.6589669897*I Test Values: {psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.21639778041374116, -0.09902593860776912] Test Values: {Rule[k, 1], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.2801868441200064, 0.09163936360272593] Test Values: {Rule[k, 2], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.11#Ex11	${\displaystyle{\displaystyle\sin\theta=(\sin\psi)/\sqrt{(1+\cos\psi)(1+\Delta(\psi))}}}$ \sin@@{\theta} = (\sin@@{\psi})/\sqrt{(1+\cos@@{\psi})(1+\Delta(\psi))}		sin(theta) = (sin(psi))/(sqrt((1 + cos(psi))*(1 + Delta(psi))))	Sin[\[Theta]] == (Sin[\[Psi]])/(Sqrt[(1 + Cos[\[Psi]])*(1 + \[CapitalDelta][\[Psi]])])	Failure	Failure	Failed [300 / 300] Result: .3459933254+.2199626413*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I} Result: -1.183718368+.7410028953*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.13267626462165183, 0.09545710280323466] Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.6075958421397494, -0.12937331954381406] Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.11#Ex12	${\displaystyle{\displaystyle\cos\theta=\sqrt{\frac{(\cos\psi)+\Delta(\psi)}{1+\Delta(\psi)}}}}$ \cos@@{\theta} = \sqrt{\frac{(\cos@@{\psi})+\Delta(\psi)}{1+\Delta(\psi)}}		cos(theta) = sqrt(((cos(psi))+ Delta(psi))/(1 + Delta(psi)))	Cos[\[Theta]] == Sqrt[Divide[(Cos[\[Psi]])+ \[CapitalDelta][\[Psi]],1 + \[CapitalDelta][\[Psi]]]]	Failure	Failure	Failed [300 / 300] Result: -.1386531520-.3275237699*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I} Result: .3585693461+.5385011568*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.027928525698177165, -0.06433717895055871] Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.11337825659380207, -0.16573354274294425] Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.11#Ex13	${\displaystyle{\displaystyle\tan\theta=\tan\left(\tfrac{1}{2}\psi\right)\sqrt{\frac{1+\cos\psi}{(\cos\psi)+\Delta(\psi)}}}}$ \tan@@{\theta} = \tan@{\tfrac{1}{2}\psi}\sqrt{\frac{1+\cos@@{\psi}}{(\cos@@{\psi})+\Delta(\psi)}}		tan(theta) = tan((1)/(2)*psi)*sqrt((1 + cos(psi))/((cos(psi))+ Delta(psi)))	Tan[\[Theta]] == Tan[Divide[1,2]*\[Psi]]*Sqrt[Divide[1 + Cos[\[Psi]],(Cos[\[Psi]])+ \[CapitalDelta][\[Psi]]]]	Failure	Failure	Failed [300 / 300] Result: .1382279959+.6687205345*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I} Result: -.8192630216+.6110829935*I Test Values: {Delta = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.12433851209893465, 0.1415108829927562] Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.5756669065605976, -0.05657247148971478] Test Values: {Rule[Δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.11.E16	${\displaystyle{\displaystyle\Pi\left(\psi,\alpha^{2},k\right)=2\Pi\left(\theta,\alpha^{2},k\right)+\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right)}}$ \incellintPik@{\psi}{\alpha^{2}}{k} = 2\incellintPik@{\theta}{\alpha^{2}}{k}+\alpha^{2}\CarlsonellintRC@{\gamma-\delta}{\gamma}		Error	EllipticPi[\[Alpha]^(2), \[Psi],(k)^2] == 2*EllipticPi[\[Alpha]^(2), \[Theta],(k)^2]+ \[Alpha]^(2)* 1/Sqrt[(((Csc[\[Theta]])^(2))- \[Alpha]^(2))^(2)*(((Csc[\[Psi]])^(2))- \[Alpha]^(2))]*Hypergeometric2F1[1/2,1/2,3/2,1-(((((Csc[\[Theta]])^(2))- \[Alpha]^(2))^(2)*(((Csc[\[Psi]])^(2))- \[Alpha]^(2)))- \[Delta])/((((Csc[\[Theta]])^(2))- \[Alpha]^(2))^(2)*(((Csc[\[Psi]])^(2))- \[Alpha]^(2)))]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Complex[-0.6318505653554005, -0.11296244472006367] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.5728350059366992, -0.1614996009729338] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[δ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.12.E1	${\displaystyle{\displaystyle K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)\right)}}$ \compellintKk@{k} = \sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)\right)	${\displaystyle{\displaystyle 0<|k^{\prime}|,|k^{\prime}|<1}}$	EllipticK(k) = sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(sqrt(1 - (k)^(2)))^(2*m)*(ln((1)/(sqrt(1 - (k)^(2))))+ d(m)), m = 0..infinity)	EllipticK[(k)^2] == Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(Sqrt[1 - (k)^(2)])^(2*m)*(Log[Divide[1,Sqrt[1 - (k)^(2)]]]+ d[m]), {m, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Skip - No test values generated
19.12.E2	${\displaystyle{\displaystyle E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2\right)_{m}}m!}{k^{\prime}}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)-\frac{1}{(2m+1)(2m+2)}\right)}}$ \compellintEk@{k} = 1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{3}{2}}{m}}{\Pochhammersym{2}{m}m!}{k^{\prime}}^{2m+2}\*\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)-\frac{1}{(2m+1)(2m+2)}\right)	${\displaystyle{\displaystyle|k^{\prime}|<1}}$	EllipticE(k) = 1 +(1)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((3)/(2), m))/(pochhammer(2, m)*factorial(m))*(sqrt(1 - (k)^(2)))^(2*m + 2)*(ln((1)/(sqrt(1 - (k)^(2))))+ d(m)-(1)/((2*m + 1)*(2*m + 2))), m = 0..infinity)	EllipticE[(k)^2] == 1 +Divide[1,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[3,2], m],Pochhammer[2, m]*(m)!]*(Sqrt[1 - (k)^(2)])^(2*m + 2)*(Log[Divide[1,Sqrt[1 - (k)^(2)]]]+ d[m]-Divide[1,(2*m + 1)*(2*m + 2)]), {m, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Failed [10 / 10] Result: Indeterminate Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[k, 1]} Result: Indeterminate Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Rule[k, 1]} ... skip entries to safe data
19.12#Ex1	${\displaystyle{\displaystyle d(m)=\psi\left(1+m\right)-\psi\left(\tfrac{1}{2}+m\right)}}$ d(m) = \digamma@{1+m}-\digamma@{\tfrac{1}{2}+m}		d(m) = Psi(1 + m)- Psi((1)/(2)+ m)	d[m] == PolyGamma[1 + m]- PolyGamma[Divide[1,2]+ m]	Failure	Failure	Failed [30 / 30] Result: .4797310429+.5000000000*I Test Values: {d = 1/2*3^(1/2)+1/2*I, m = 1} Result: 1.512423114+1.*I Test Values: {d = 1/2*3^(1/2)+1/2*I, m = 2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[0.04671834077232884, 0.24999999999999997] Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[m, 1]} Result: Complex[0.6463977093312149, 0.49999999999999994] Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[m, 2]} ... skip entries to safe data
19.12#Ex2	${\displaystyle{\displaystyle d(m+1)=d(m)-\frac{2}{(2m+1)(2m+2)}}}$ d(m+1) = d(m)-\frac{2}{(2m+1)(2m+2)}		d(m + 1) = d(m)-(2)/((2*m + 1)*(2*m + 2))	d[m + 1] == d[m]-Divide[2,(2*m + 1)*(2*m + 2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.14.E1	${\displaystyle{\displaystyle\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{t^{3}-1}}=3^{-1/4}F\left(\phi,k\right)}}$ \int_{1}^{x}\frac{\diff{t}}{\sqrt{t^{3}-1}} = 3^{-1/4}\incellintFk@{\phi}{k}	${\displaystyle{\displaystyle\cos\phi=\dfrac{\sqrt{3}+1-x}{\sqrt{3}-1+x},k^{2}=\dfrac{2-\sqrt{3}}{4}}}$	int((1)/(sqrt((t)^(3)- 1)), t = 1..x) = (3)^(- 1/4)* EllipticF(sin(phi), k)	Integrate[Divide[1,Sqrt[(t)^(3)- 1]], {t, 1, x}, GenerateConditions->None] == (3)^(- 1/4)* EllipticF[\[Phi], (k)^2]	Failure	Aborted	Error	Skipped - Because timed out
19.14.E2	${\displaystyle{\displaystyle\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{3}}}=3^{-1/4}F\left(\phi,k\right)}}$ \int_{x}^{1}\frac{\diff{t}}{\sqrt{1-t^{3}}} = 3^{-1/4}\incellintFk@{\phi}{k}	${\displaystyle{\displaystyle\cos\phi=\dfrac{\sqrt{3}-1+x}{\sqrt{3}+1-x},k^{2}=\dfrac{2+\sqrt{3}}{4}}}$	int((1)/(sqrt(1 - (t)^(3))), t = x..1) = (3)^(- 1/4)* EllipticF(sin(phi), k)	Integrate[Divide[1,Sqrt[1 - (t)^(3)]], {t, x, 1}, GenerateConditions->None] == (3)^(- 1/4)* EllipticF[\[Phi], (k)^2]	Failure	Aborted	Error	Skip - No test values generated
19.14.E3	${\displaystyle{\displaystyle\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{4}}}=\frac{\operatorname{sign}\left(x\right)}{2}F\left(\phi,k\right)}}$ \int_{0}^{x}\frac{\diff{t}}{\sqrt{1+t^{4}}} = \frac{\sign@{x}}{2}\incellintFk@{\phi}{k}	${\displaystyle{\displaystyle\cos\phi=\dfrac{1-x^{2}}{1+x^{2}},k^{2}=\dfrac{1}{2}}}$	int((1)/(sqrt(1 + (t)^(4))), t = 0..x) = (signum(x))/(2)*EllipticF(sin(phi), k)	Integrate[Divide[1,Sqrt[1 + (t)^(4)]], {t, 0, x}, GenerateConditions->None] == Divide[Sign[x],2]*EllipticF[\[Phi], (k)^2]	Failure	Failure	Error	Skip - No test values generated
19.14.E4	${\displaystyle{\displaystyle\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}}=\frac{1}{\sqrt{\gamma-\alpha}}F\left(\phi,k\right)}}$ \int_{y}^{x}\frac{\diff{t}}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}} = \frac{1}{\sqrt{\gamma-\alpha}}\incellintFk@{\phi}{k}	${\displaystyle{\displaystyle k^{2}=\ifrac{(\gamma-\beta)}{(\gamma-\alpha)}}}$	int((1)/(sqrt((a[1]+ b[1]*(t)^(2))*(a[2]+ b[2]*(t)^(2)))), t = y..x) = (1)/(sqrt(gamma - alpha))*EllipticF(sin(phi), k)	Integrate[Divide[1,Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(t)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(t)^(2))]], {t, y, x}, GenerateConditions->None] == Divide[1,Sqrt[\[Gamma]- \[Alpha]]]*EllipticF[\[Phi], (k)^2]	Skipped - Unable to analyze test case: Null	Skipped - Unable to analyze test case: Null	-	-
19.14.E5	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{\gamma-\alpha}{U^{2}+\gamma}}}$ \sin^{2}@@{\phi} = \frac{\gamma-\alpha}{U^{2}+\gamma}		(sin(phi))^(2) = (gamma - alpha)/((U)^(2)+ gamma)	(Sin[\[Phi]])^(2) == Divide[\[Gamma]- \[Alpha],(U)^(2)+ \[Gamma]]	Failure	Failure	Failed [300 / 300] Result: 1.144207228+.1616580578*I Test Values: {U = 1/2*3^(1/2)+1/2*I, alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I} Result: .2329549284-1.570148532*I Test Values: {U = 1/2*3^(1/2)+1/2*I, alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.0397570908067482, -1.0061601508735134] Test Values: {Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.7911458419033055, -1.4391726141222814] Test Values: {Rule[U, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.14.E6	${\displaystyle{\displaystyle(x^{2}-y^{2})U=x\sqrt{(a_{1}+b_{1}y^{2})(a_{2}+b_{2}y^{2})}+y\sqrt{(a_{1}+b_{1}x^{2})(a_{2}+b_{2}x^{2})}}}$ (x^{2}-y^{2})U = x\sqrt{(a_{1}+b_{1}y^{2})(a_{2}+b_{2}y^{2})}+y\sqrt{(a_{1}+b_{1}x^{2})(a_{2}+b_{2}x^{2})}		((x)^(2)- (y)^(2))*U = x*sqrt((a[1]+ b[1]*(y)^(2))*(a[2]+ b[2]*(y)^(2)))+ y*sqrt((a[1]+ b[1]*(x)^(2))*(a[2]+ b[2]*(x)^(2)))	((x)^(2)- (y)^(2))*U == x*Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(y)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(y)^(2))]+ y*Sqrt[(Subscript[a, 1]+ Subscript[b, 1]*(x)^(2))*(Subscript[a, 2]+ Subscript[b, 2]*(x)^(2))]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.14.E7	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}}}$ \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)x^{2}}{a_{1}a_{2}+\gamma x^{2}}		(sin(phi))^(2) = ((gamma - alpha)*(x)^(2))/(a[1]*a[2]+ gamma*(x)^(2))	(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*(x)^(2),Subscript[a, 1]*Subscript[a, 2]+ \[Gamma]*(x)^(2)]	Failure	Failure	Failed [300 / 300] Result: 1.560947444+.1288116535*I Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, a[1] = 1/2*3^(1/2)+1/2*I, a[2] = 1/2*3^(1/2)+1/2*I} Result: 2.678639127-1.794319469*I Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, x = 3/2, a[1] = 1/2*3^(1/2)+1/2*I, a[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.3471528039744003, -1.172411794219179] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.5030688086095803, -1.7852795940180226] Test Values: {Rule[x, 1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[a, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.14.E8	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}}}$ \sin^{2}@@{\phi} = \frac{\gamma-\alpha}{b_{1}b_{2}y^{2}+\gamma}		(sin(phi))^(2) = (gamma - alpha)/(b[1]*b[2]*(y)^(2)+ gamma)	(Sin[\[Phi]])^(2) == Divide[\[Gamma]- \[Alpha],Subscript[b, 1]*Subscript[b, 2]*(y)^(2)+ \[Gamma]]	Failure	Failure	Failed [300 / 300] Result: .8585159693+.3113806358*I Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, y = -3/2, b[1] = 1/2*3^(1/2)+1/2*I, b[2] = 1/2*3^(1/2)+1/2*I} Result: .2216600130+.2500138214*I Test Values: {alpha = 3/2, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, y = -3/2, b[1] = 1/2*3^(1/2)+1/2*I, b[2] = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.683185473382228, -0.7175596041712626] Test Values: {Rule[y, -1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.5335538340604822, -1.7418837307419275] Test Values: {Rule[y, -1.5], Rule[α, 1.5], Rule[γ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[b, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.14.E9	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a_{2}+b_{2}x^{2})}}}$ \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(x^{2}-y^{2})}{\gamma(x^{2}-y^{2})-a_{1}(a_{2}+b_{2}x^{2})}		(sin(phi))^(2) = ((gamma - alpha)*((x)^(2)- (y)^(2)))/(gamma*((x)^(2)- (y)^(2))- a[1]*(a[2]+ b[2]*(x)^(2)))	(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*((x)^(2)- (y)^(2)),\[Gamma]*((x)^(2)- (y)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(x)^(2))]	Failure	Failure	Manual Skip!	Skipped - Because timed out
19.14.E10	${\displaystyle{\displaystyle{\sin^{2}}\phi=\frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a_{2}+b_{2}y^{2})}}}$ \sin^{2}@@{\phi} = \frac{(\gamma-\alpha)(y^{2}-x^{2})}{\gamma(y^{2}-x^{2})-a_{1}(a_{2}+b_{2}y^{2})}		(sin(phi))^(2) = ((gamma - alpha)*((y)^(2)- (x)^(2)))/(gamma*((y)^(2)- (x)^(2))- a[1]*(a[2]+ b[2]*(y)^(2)))	(Sin[\[Phi]])^(2) == Divide[(\[Gamma]- \[Alpha])*((y)^(2)- (x)^(2)),\[Gamma]*((y)^(2)- (x)^(2))- Subscript[a, 1]*(Subscript[a, 2]+ Subscript[b, 2]*(y)^(2))]	Failure	Failure	Manual Skip!	Skipped - Because timed out
19.16.E1	${\displaystyle{\displaystyle R_{F}\left(x,y,z\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)}}}$ \CarlsonsymellintRF@{x}{y}{z} = \frac{1}{2}\int_{0}^{\infty}\frac{\diff{t}}{s(t)}		0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = (1)/(2)*int((1)/(s(t)), t = 0..infinity)	EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == Divide[1,2]*Integrate[Divide[1,s[t]], {t, 0, Infinity}, GenerateConditions->None]	Failure	Failure	Failed [108 / 108] Result: Float(infinity)+1.326265449*I Test Values: {s = -3/2, x = 3/2, y = -3/2} Result: Float(infinity)+Float(infinity)*I Test Values: {s = -3/2, x = 3/2, y = 3/2} ... skip entries to safe data	Failed [108 / 108] Result: Complex[52.57956240437182, 0.6784437678906974] Test Values: {Rule[s, -1.5], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[52.453473067488765, -0.7809212115368181] Test Values: {Rule[s, -1.5], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.16.E2	${\displaystyle{\displaystyle R_{J}\left(x,y,z,p\right)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+p)}}}$ \CarlsonsymellintRJ@{x}{y}{z}{p} = \frac{3}{2}\int_{0}^{\infty}\frac{\diff{t}}{s(t)(t+p)}		Error	3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == Divide[3,2]*Integrate[Divide[1,s[t]*(t + p)], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
19.16.E3	${\displaystyle{\displaystyle R_{G}\left(x,y,z\right)=\frac{1}{4\pi}\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\left(x{\sin^{2}}\theta{\cos^{2}}\phi+y{\sin^{2}}\theta{\sin^{2}}\phi+z{\cos^{2}}\theta\right)^{\frac{1}{2}}\sin\theta\mathrm{d}\theta\mathrm{d}\phi}}$ \CarlsonsymellintRG@{x}{y}{z} = \frac{1}{4\pi}\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\left(x\sin^{2}@@{\theta}\cos^{2}@@{\phi}+y\sin^{2}@@{\theta}\sin^{2}@@{\phi}+z\cos^{2}@@{\theta}\right)^{\frac{1}{2}}\sin@@{\theta}\diff{\theta}\diff{\phi}		Error	Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) == Divide[1,4*Pi]*Integrate[Integrate[(x*(Sin[\[Theta]])^(2)* (Cos[\[Phi]])^(2)+ y*(Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2)+(x + y*I)*(Cos[\[Theta]])^(2))^(Divide[1,2])* Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None], {\[Phi], 0, 2*Pi}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.16.E4	${\displaystyle{\displaystyle s(t)=\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}}$ s(t) = \sqrt{t+x}\sqrt{t+y}\sqrt{t+z}		s(t) = sqrt(t + x)*sqrt(t + y)*sqrt(t +(x + y*I))	s[t] == Sqrt[t + x]*Sqrt[t + y]*Sqrt[t +(x + y*I)]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.16.E5	${\displaystyle{\displaystyle R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)}}$ \CarlsonsymellintRD@{x}{y}{z} = \CarlsonsymellintRJ@{x}{y}{z}{z}		Error	3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2)) == 3*(x + y*I-x)/(x + y*I-x + y*I)*(EllipticPi[(x + y*I-x + y*I)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[0.37100270206594405, -0.09129381935817127] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[0.5182279531589904, 0.0513630200054771] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.16.E5	${\displaystyle{\displaystyle R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+z)}}}$ \CarlsonsymellintRJ@{x}{y}{z}{z} = \frac{3}{2}\int_{0}^{\infty}\frac{\diff{t}}{s(t)(t+z)}		Error	3*(x + y*I-x)/(x + y*I-x + y*I)*(EllipticPi[(x + y*I-x + y*I)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == Divide[3,2]*Integrate[Divide[1,s[t]*(t +(x + y*I))], {t, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Skipped - Because timed out
19.16.E6	${\displaystyle{\displaystyle R_{C}\left(x,y\right)=R_{F}\left(x,y,y\right)}}$ \CarlsonellintRC@{x}{y} = \CarlsonsymellintRF@{x}{y}{y}		Error	1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)] == EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]/Sqrt[y-x]	Missing Macro Error	Failure	-	Failed [3 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} Result: Indeterminate Test Values: {Rule[x, 0.5], Rule[y, 0.5]} ... skip entries to safe data
19.16#Ex3	${\displaystyle{\displaystyle c={\csc^{2}}\phi}}$ c = \csc^{2}@@{\phi}		c = (csc(phi))^(2)	c == (Csc[\[Phi]])^(2)	Failure	Failure	Failed [60 / 60] Result: -2.359812877+.7993130071*I Test Values: {c = -3/2, phi = 1/2*3^(1/2)+1/2*I} Result: -1.296085040-.8173084059*I Test Values: {c = -3/2, phi = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [60 / 60] Result: Complex[-3.841312467237177, 3.4490957612740374] Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.17530792640393877, -3.4502399957777015] Test Values: {Rule[c, -1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.18.E1	${\displaystyle{\displaystyle\frac{\partial R_{F}\left(x,y,z\right)}{\partial z}=-\tfrac{1}{6}R_{D}\left(x,y,z\right)}}$ \pderiv{\CarlsonsymellintRF@{x}{y}{z}}{z} = -\tfrac{1}{6}\CarlsonsymellintRD@{x}{y}{z}		Error	(D[EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x], {temp, 1}]/.temp-> (x + y*I)) == -Divide[1,6]*3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2))	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[0.03790163875178684, -0.07848225754688502] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[0.07302626282106058, 0.09607801553820669] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.18.E2	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x+c\right)=\tfrac{1}{2}R_{F}\left(x+a,x+b,x+c\right)}}$ \deriv{}{x}\CarlsonsymellintRG@{x+a}{x+b}{x+c} = \tfrac{1}{2}\CarlsonsymellintRF@{x+a}{x+b}{x+c}		Error	D[Sqrt[x + c-x + a]*(EllipticE[ArcCos[Sqrt[x + a/x + c]],(x + c-x + b)/(x + c-x + a)]+(Cot[ArcCos[Sqrt[x + a/x + c]]])^2*EllipticF[ArcCos[Sqrt[x + a/x + c]],(x + c-x + b)/(x + c-x + a)]+Cot[ArcCos[Sqrt[x + a/x + c]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x + a/x + c]]]^2]), x] == Divide[1,2]*EllipticF[ArcCos[Sqrt[x + a/x + c]],(x + c-x + b)/(x + c-x + a)]/Sqrt[x + c-x + a]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Plus[Complex[0.4534498410585545, 0.2544306388611797], Times[Complex[0.0, 1.7320508075688772], Plus[Complex[-0.5166444818917079, -0.6544984694978735], Times[Complex[0.0, 0.5892556509887895], Power[k, 2], Power[Plus[1.0, Times[-2.0, Power[k, 2]]], Rational[-1, 2]]], Times[Complex[0.0, -0.29462782549439476], Power[Plus[1.0, Times[-2.0, Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 1.5]} Result: Plus[Complex[0.4534498410585545, 0.1389138883676965], Times[Complex[0.0, 1.7320508075688772], Plus[Complex[-1.7435577900831345, -0.43982297150257077], Times[Complex[0.0, 3.1304951684997055], Power[k, 2], Power[Plus[1.0, Times[-5.0, Power[k, 2]]], Rational[-1, 2]]], Times[Complex[0.0, -0.15652475842498526], Power[Plus[1.0, Times[-5.0, Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 0.5]} ... skip entries to safe data
19.18.E9	${\displaystyle{\displaystyle\left(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{\partial}{\partial z}\right)R_{F}\left(x,y,z\right)=-\tfrac{1}{2}R_{F}\left(x,y,z\right)}}$ \left(x\pderiv{}{x}+y\pderiv{}{y}+z\pderiv{}{z}\right)\CarlsonsymellintRF@{x}{y}{z} = -\tfrac{1}{2}\CarlsonsymellintRF@{x}{y}{z}		(x*diff(+ y*diff(+subs( temp=(x + y*I), diff( temp, temp$(1) ) ), y), x))*0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = -(1)/(2)*0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)	(x*D[+ y*D[+(D[temp, {temp, 1}]/.temp-> (x + y*I)), y], x])*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == -Divide[1,2]*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]	Aborted	Failure	Failed [18 / 18] Result: -.8633499928+.6631327246*I Test Values: {x = 3/2, y = -3/2} Result: Float(infinity)+Float(infinity)*I Test Values: {x = 3/2, y = 3/2} ... skip entries to safe data	Failed [18 / 18] Result: Complex[-0.08107235486578032, 0.3392218839453487] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-0.14411702330731, -0.3904606057684091] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.18.E14	${\displaystyle{\displaystyle\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{2}}+\frac{1}{y}\frac{\partial w}{\partial y}}}$ \pderiv[2]{w}{x} = \pderiv[2]{w}{y}+\frac{1}{y}\pderiv{w}{y}		diff(w, [x$(2)]) = diff(w, [y$(2)])+(1)/(y)*diff(w, y)	D[w, {x, 2}] == D[w, {y, 2}]+Divide[1,y]*D[w, y]	Successful	Successful	-	Successful [Tested: 180]
19.18.E15	${\displaystyle{\displaystyle\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}}}$ \pderiv[2]{W}{t} = \pderiv[2]{W}{x}+\pderiv[2]{W}{y}		diff(W, [t$(2)]) = diff(W, [x$(2)])+ diff(W, [y$(2)])	D[W, {t, 2}] == D[W, {x, 2}]+ D[W, {y, 2}]	Successful	Successful	-	Successful [Tested: 300]
19.18.E16	${\displaystyle{\displaystyle\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0}}$ \pderiv[2]{u}{x}+\pderiv[2]{u}{y}+\frac{1}{y}\pderiv{u}{y} = 0		diff(u, [x$(2)])+ diff(u, [y$(2)])+(1)/(y)*diff(u, y) = 0	D[u, {x, 2}]+ D[u, {y, 2}]+Divide[1,y]*D[u, y] == 0	Successful	Successful	-	Successful [Tested: 180]
19.18.E17	${\displaystyle{\displaystyle\frac{{\partial}^{2}U}{{\partial x}^{2}}+\frac{{\partial}^{2}U}{{\partial y}^{2}}+\frac{{\partial}^{2}U}{{\partial z}^{2}}=0}}$ \pderiv[2]{U}{x}+\pderiv[2]{U}{y}+\pderiv[2]{U}{z} = 0		diff(U, [x$(2)])+ diff(U, [y$(2)])+ subs( temp=(x + y*I), diff( U, temp$(2) ) ) = 0	D[U, {x, 2}]+ D[U, {y, 2}]+ (D[U, {temp, 2}]/.temp-> (x + y*I)) == 0	Successful	Successful	-	Successful [Tested: 180]
19.19#Ex1	${\displaystyle{\displaystyle A=\frac{1}{n}\sum_{j=1}^{n}z_{j}}}$ A = \frac{1}{n}\sum_{j=1}^{n}z_{j}		A = (1)/(n)*sum(z[j], j = 1..n)	A == Divide[1,n]*Sum[Subscript[z, j], {j, 1, n}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.19#Ex2	${\displaystyle{\displaystyle Z_{j}=1-(z_{j}/A)}}$ Z_{j} = 1-(z_{j}/A)		Z[j] = 1 -(z[j]/A)	Subscript[Z, j] == 1 -(Subscript[z, j]/A)	Skipped - no semantic math	Skipped - no semantic math	-	-
19.19#Ex3	${\displaystyle{\displaystyle E_{1}(\mathbf{Z})=0}}$ E_{1}(\mathbf{Z}) = 0	${\displaystyle{\displaystyle|Z_{j}|<1}}$	E[1](Z) = 0	Subscript[E, 1][Z] == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
19.20#Ex1	${\displaystyle{\displaystyle R_{F}\left(x,x,x\right)=x^{-1/2}}}$ \CarlsonsymellintRF@{x}{x}{x} = x^{-1/2}		0.5*int(1/(sqrt(t+x)*sqrt(t+x)*sqrt(t+x)), t = 0..infinity) = (x)^(- 1/2)	EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]/Sqrt[x-x] == (x)^(- 1/2)	Failure	Failure	Successful [Tested: 3]	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[x, 0.5]} ... skip entries to safe data
19.20#Ex2	${\displaystyle{\displaystyle R_{F}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{-1/2}R_{F}\left(x,y,z\right)}}$ \CarlsonsymellintRF@{\lambda x}{\lambda y}{\lambda z} = \lambda^{-1/2}\CarlsonsymellintRF@{x}{y}{z}		0.5*int(1/(sqrt(t+lambda*x)*sqrt(t+lambda*y)*sqrt(t+lambda*(x + y*I))), t = 0..infinity) = (lambda)^(- 1/2)* 0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)	EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]/Sqrt[\[Lambda]*(x + y*I)-\[Lambda]*x] == \[Lambda]^(- 1/2)* EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]	Aborted	Failure	Skipped - Because timed out	Failed [180 / 180] Result: Complex[-0.15259412278903736, 0.06775202977854555] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.05999241929777854, 0.15580825868890358] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20#Ex3	${\displaystyle{\displaystyle R_{F}\left(x,y,y\right)=R_{C}\left(x,y\right)}}$ \CarlsonsymellintRF@{x}{y}{y} = \CarlsonellintRC@{x}{y}		Error	EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]/Sqrt[y-x] == 1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]	Missing Macro Error	Failure	-	Failed [3 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} Result: Indeterminate Test Values: {Rule[x, 0.5], Rule[y, 0.5]} ... skip entries to safe data
19.20#Ex4	${\displaystyle{\displaystyle R_{F}\left(0,y,y\right)=\tfrac{1}{2}\pi y^{-1/2}}}$ \CarlsonsymellintRF@{0}{y}{y} = \tfrac{1}{2}\pi y^{-1/2}		0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+y)), t = 0..infinity) = (1)/(2)*Pi*(y)^(- 1/2)	EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)]/Sqrt[y-0] == Divide[1,2]*Pi*(y)^(- 1/2)	Failure	Successful	Failed [3 / 6] Result: 2.565099660*I Test Values: {y = -3/2} Result: 4.442882938*I Test Values: {y = -1/2} ... skip entries to safe data	Successful [Tested: 6]
19.20#Ex5	${\displaystyle{\displaystyle R_{F}\left(0,0,z\right)=\infty}}$ \CarlsonsymellintRF@{0}{0}{z} = \infty		0.5*int(1/(sqrt(t+0)*sqrt(t+0)*sqrt(t+z)), t = 0..infinity) = infinity	EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]/Sqrt[z-0] == Infinity	Failure	Failure	Skipped - Because timed out	Failed [7 / 7] Result: Indeterminate Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20.E2	${\displaystyle{\displaystyle\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{F}\left(0,1,2\right)}}$ \int_{0}^{1}\frac{\diff{t}}{\sqrt{1-t^{4}}} = \CarlsonsymellintRF@{0}{1}{2}		int((1)/(sqrt(1 - (t)^(4))), t = 0..1) = 0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity)	Integrate[Divide[1,Sqrt[1 - (t)^(4)]], {t, 0, 1}, GenerateConditions->None] == EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0]	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]
19.20.E2	${\displaystyle{\displaystyle R_{F}\left(0,1,2\right)=\frac{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}}}$ \CarlsonsymellintRF@{0}{1}{2} = \frac{\left(\EulerGamma@{\frac{1}{4}}\right)^{2}}{4(2\pi)^{1/2}}		0.5*int(1/(sqrt(t+0)*sqrt(t+1)*sqrt(t+2)), t = 0..infinity) = ((GAMMA((1)/(4)))^(2))/(4*(2*Pi)^(1/2))	EllipticF[ArcCos[Sqrt[0/2]],(2-1)/(2-0)]/Sqrt[2-0] == Divide[(Gamma[Divide[1,4]])^(2),4*(2*Pi)^(1/2)]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 1]
19.20.E2	${\displaystyle{\displaystyle\frac{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(2\pi)^{1/2}}=1.31102\;87771\;46059\;90523\;\dots}}$ \frac{\left(\EulerGamma@{\frac{1}{4}}\right)^{2}}{4(2\pi)^{1/2}} = 1.31102\;87771\;46059\;90523\;\dots		((GAMMA((1)/(4)))^(2))/(4*(2*Pi)^(1/2)) = 1.31102877714605990523	Divide[(Gamma[Divide[1,4]])^(2),4*(2*Pi)^(1/2)] == 1.31102877714605990523	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]
19.20#Ex6	${\displaystyle{\displaystyle R_{G}\left(x,x,x\right)=x^{1/2}}}$ \CarlsonsymellintRG@{x}{x}{x} = x^{1/2}		Error	Sqrt[x-x]*(EllipticE[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]+(Cot[ArcCos[Sqrt[x/x]]])^2*EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]+Cot[ArcCos[Sqrt[x/x]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x]]]^2]) == (x)^(1/2)	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[x, 0.5]} ... skip entries to safe data
19.20#Ex7	${\displaystyle{\displaystyle R_{G}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{1/2}R_{G}\left(x,y,z\right)}}$ \CarlsonsymellintRG@{\lambda x}{\lambda y}{\lambda z} = \lambda^{1/2}\CarlsonsymellintRG@{x}{y}{z}		Error	Sqrt[\[Lambda]*(x + y*I)-\[Lambda]*x]*(EllipticE[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]+(Cot[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]]])^2*EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]+Cot[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]]]^2]) == \[Lambda]^(1/2)* Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2])	Missing Macro Error	Aborted	-	Failed [180 / 180] Result: Plus[Times[Complex[-0.75, 0.4330127018922193], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], Times[Complex[0.75, -0.43301270189221935], Plus[Complex[0.469094970899074, 0.7900882534928779], Times[Complex[0.1542171038749957, -1.1011185950707625], Power[Plus[1.0, Times[Complex[1.25, -2.25], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Times[Complex[-0.8365163037378078, -0.22414386804201336], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]], Times[Complex[0.8365163037378078, 0.22414386804201325], Plus[Complex[0.46909497089907387, 0.7900882534928779], Times[Complex[0.1542171038749957, -1.1011185950707625], Power[Plus[1.0, Times[Complex[1.25, -2.25], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20#Ex8	${\displaystyle{\displaystyle R_{G}\left(0,y,y\right)=\tfrac{1}{4}\pi y^{1/2}}}$ \CarlsonsymellintRG@{0}{y}{y} = \tfrac{1}{4}\pi y^{1/2}		Error	Sqrt[y-0]*(EllipticE[ArcCos[Sqrt[0/y]],(y-y)/(y-0)]+(Cot[ArcCos[Sqrt[0/y]]])^2*EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)]+Cot[ArcCos[Sqrt[0/y]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/y]]]^2]) == Divide[1,4]*Pi*(y)^(1/2)	Missing Macro Error	Failure	-	Failed [6 / 6] Result: Complex[0.0, 0.961912372621398] Test Values: {Rule[y, -1.5]} Result: 0.961912372621398 Test Values: {Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex9	${\displaystyle{\displaystyle R_{G}\left(0,0,z\right)=\tfrac{1}{2}z^{1/2}}}$ \CarlsonsymellintRG@{0}{0}{z} = \tfrac{1}{2}z^{1/2}		Error	Sqrt[z-0]*(EllipticE[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]+(Cot[ArcCos[Sqrt[0/z]]])^2*EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]+Cot[ArcCos[Sqrt[0/z]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/z]]]^2]) == Divide[1,2]*(z)^(1/2)	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Indeterminate Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20.E5	${\displaystyle{\displaystyle 2R_{G}\left(x,y,y\right)=yR_{C}\left(x,y\right)+\sqrt{x}}}$ 2\CarlsonsymellintRG@{x}{y}{y} = y\CarlsonellintRC@{x}{y}+\sqrt{x}		Error	2*Sqrt[y-x]*(EllipticE[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]+(Cot[ArcCos[Sqrt[x/y]]])^2*EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]+Cot[ArcCos[Sqrt[x/y]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/y]]]^2]) == y*1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]+Sqrt[x]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Plus[Complex[-1.988036787975128, -1.360349523175663], Times[Complex[0.0, 3.4641016151377544], Plus[Complex[0.7853981633974483, -0.44068679350977147], Times[Complex[0.0, 0.7071067811865475], Power[Plus[1.0, Times[-2.0, Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex10	${\displaystyle{\displaystyle R_{J}\left(x,x,x,x\right)=x^{-3/2}}}$ \CarlsonsymellintRJ@{x}{x}{x}{x} = x^{-3/2}		Error	3*(x-x)/(x-x)*(EllipticPi[(x-x)/(x-x),ArcCos[Sqrt[x/x]],(x-x)/(x-x)]-EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)])/Sqrt[x-x] == (x)^(- 3/2)	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[x, 0.5]} ... skip entries to safe data
19.20#Ex11	${\displaystyle{\displaystyle R_{J}\left(\lambda x,\lambda y,\lambda z,\lambda p\right)=\lambda^{-3/2}R_{J}\left(x,y,z,p\right)}}$ \CarlsonsymellintRJ@{\lambda x}{\lambda y}{\lambda z}{\lambda p} = \lambda^{-3/2}\CarlsonsymellintRJ@{x}{y}{z}{p}		Error	3*(\[Lambda]*(x + y*I)-\[Lambda]*x)/(\[Lambda]*(x + y*I)-\[Lambda]*p)*(EllipticPi[(\[Lambda]*(x + y*I)-\[Lambda]*p)/(\[Lambda]*(x + y*I)-\[Lambda]*x),ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]-EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)])/Sqrt[\[Lambda]*(x + y*I)-\[Lambda]*x] == \[Lambda]^(- 3/2)* 3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[0.8261798979421457, -0.5239696989052641] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.5256524914787406, -1.066611458671583] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20#Ex12	${\displaystyle{\displaystyle R_{J}\left(x,y,z,z\right)=R_{D}\left(x,y,z\right)}}$ \CarlsonsymellintRJ@{x}{y}{z}{z} = \CarlsonsymellintRD@{x}{y}{z}		Error	3*(x + y*I-x)/(x + y*I-x + y*I)*(EllipticPi[(x + y*I-x + y*I)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == 3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2))	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[-0.37100270206594405, 0.09129381935817127] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-0.5182279531589904, -0.0513630200054771] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex13	${\displaystyle{\displaystyle R_{J}\left(0,0,z,p\right)=\infty}}$ \CarlsonsymellintRJ@{0}{0}{z}{p} = \infty		Error	3*(z-0)/(z-p)*(EllipticPi[(z-p)/(z-0),ArcCos[Sqrt[0/z]],(z-0)/(z-0)]-EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)])/Sqrt[z-0] == Infinity	Missing Macro Error	Failure	-	Failed [70 / 70] Result: Indeterminate Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20#Ex14	${\displaystyle{\displaystyle R_{J}\left(x,x,x,p\right)=R_{D}\left(p,p,x\right)}}$ \CarlsonsymellintRJ@{x}{x}{x}{p} = \CarlsonsymellintRD@{p}{p}{x}	${\displaystyle{\displaystyle x\neq p,xp\neq 0}}$	Error	3*(x-x)/(x-p)*(EllipticPi[(x-p)/(x-x),ArcCos[Sqrt[x/x]],(x-x)/(x-x)]-EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)])/Sqrt[x-x] == 3*(EllipticF[ArcCos[Sqrt[p/x]],(x-p)/(x-p)]-EllipticE[ArcCos[Sqrt[p/x]],(x-p)/(x-p)])/((x-p)*(x-p)^(1/2))	Missing Macro Error	Failure	-	Failed [27 / 27] Result: Indeterminate Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 0.5]} ... skip entries to safe data
19.20#Ex14	${\displaystyle{\displaystyle R_{D}\left(p,p,x\right)=\frac{3}{x-p}\left(R_{C}\left(x,p\right)-\frac{1}{\sqrt{x}}\right)}}$ \CarlsonsymellintRD@{p}{p}{x} = \frac{3}{x-p}\left(\CarlsonellintRC@{x}{p}-\frac{1}{\sqrt{x}}\right)	${\displaystyle{\displaystyle x\neq p,xp\neq 0}}$	Error	3*(EllipticF[ArcCos[Sqrt[p/x]],(x-p)/(x-p)]-EllipticE[ArcCos[Sqrt[p/x]],(x-p)/(x-p)])/((x-p)*(x-p)^(1/2)) == Divide[3,x - p]*(1/Sqrt[p]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(p)]-Divide[1,Sqrt[x]])	Missing Macro Error	Failure	-	Failed [9 / 27] Result: Complex[1.0177225554447191, 2.220446049250313*^-16] Test Values: {Rule[p, -1.5], Rule[x, 1.5]} Result: Complex[1.1652542988181402, 6.661338147750939*^-16] Test Values: {Rule[p, -1.5], Rule[x, 0.5]} ... skip entries to safe data
19.20#Ex15	${\displaystyle{\displaystyle R_{J}\left(0,y,y,p\right)=\frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})}}}$ \CarlsonsymellintRJ@{0}{y}{y}{p} = \frac{3\pi}{2(y\sqrt{p}+p\sqrt{y})}	${\displaystyle{\displaystyle p>0}}$	Error	3*(y-0)/(y-p)*(EllipticPi[(y-p)/(y-0),ArcCos[Sqrt[0/y]],(y-y)/(y-0)]-EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)])/Sqrt[y-0] == Divide[3*Pi,2*(y*Sqrt[p]+ p*Sqrt[y])]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[-0.6412749150809316, 3.2063745754046598] Test Values: {Rule[p, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[p, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex16	${\displaystyle{\displaystyle R_{J}\left(0,y,y,-q\right)=\frac{-3\pi}{2\sqrt{y}(y+q)}}}$ \CarlsonsymellintRJ@{0}{y}{y}{-q} = \frac{-3\pi}{2\sqrt{y}(y+q)}	${\displaystyle{\displaystyle q>0}}$	Error	3*(y-0)/(y-- q)*(EllipticPi[(y-- q)/(y-0),ArcCos[Sqrt[0/y]],(y-y)/(y-0)]-EllipticF[ArcCos[Sqrt[0/y]],(y-y)/(y-0)])/Sqrt[y-0] == Divide[- 3*Pi,2*Sqrt[y]*(y + q)]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: DirectedInfinity[] Test Values: {Rule[q, 1.5], Rule[y, -1.5]} Result: Plus[1.282549830161864, Times[2.449489742783178, Plus[-1.5707963267948966, Times[1.5707963267948966, Power[Plus[1.0, Times[-1.0, Decrement[1.5]]], Rational[-1, 2]]]], Power[Decrement[1.5], -1]]] Test Values: {Rule[q, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex17	${\displaystyle{\displaystyle R_{J}\left(x,y,y,p\right)=\frac{3}{p-y}(R_{C}\left(x,y\right)-R_{C}\left(x,p\right))}}$ \CarlsonsymellintRJ@{x}{y}{y}{p} = \frac{3}{p-y}(\CarlsonellintRC@{x}{y}-\CarlsonellintRC@{x}{p})	${\displaystyle{\displaystyle p\neq y}}$	Error	3*(y-x)/(y-p)*(EllipticPi[(y-p)/(y-x),ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/Sqrt[y-x] == Divide[3,p - y]*(1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]- 1/Sqrt[p]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(p)])	Missing Macro Error	Aborted	-	Failed [157 / 162] Result: Complex[0.40904124998304914, 6.107600792054881] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex18	${\displaystyle{\displaystyle R_{J}\left(x,y,y,y\right)=R_{D}\left(x,y,y\right)}}$ \CarlsonsymellintRJ@{x}{y}{y}{y} = \CarlsonsymellintRD@{x}{y}{y}		Error	3*(y-x)/(y-y)*(EllipticPi[(y-y)/(y-x),ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/Sqrt[y-x] == 3*(EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticE[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/((y-y)*(y-x)^(1/2))	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20.E9	${\displaystyle{\displaystyle R_{J}\left(0,y,z,+\sqrt{yz}\right)=+\frac{3}{2\sqrt{yz}}R_{F}\left(0,y,z\right)}}$ \CarlsonsymellintRJ@{0}{y}{z}{+\sqrt{yz}} = +\frac{3}{2\sqrt{yz}}\CarlsonsymellintRF@{0}{y}{z}		Error	3*(x + y*I-0)/(x + y*I-+Sqrt[y*(x + y*I)])*(EllipticPi[(x + y*I-+Sqrt[y*(x + y*I)])/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == +Divide[3,2*Sqrt[y*(x + y*I)]]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[-0.9141259292931587, -0.9706303463287326] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-4.407772019377616, 0.7576222483343515] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20.E9	${\displaystyle{\displaystyle R_{J}\left(0,y,z,-\sqrt{yz}\right)=-\frac{3}{2\sqrt{yz}}R_{F}\left(0,y,z\right)}}$ \CarlsonsymellintRJ@{0}{y}{z}{-\sqrt{yz}} = -\frac{3}{2\sqrt{yz}}\CarlsonsymellintRF@{0}{y}{z}		Error	3*(x + y*I-0)/(x + y*I--Sqrt[y*(x + y*I)])*(EllipticPi[(x + y*I--Sqrt[y*(x + y*I)])/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == -Divide[3,2*Sqrt[y*(x + y*I)]]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[0.1671030668705316, -0.09828926199489627] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-0.7387931095854892, 1.0731895314108653] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex19	${\displaystyle{\displaystyle\lim_{p\to 0+}\sqrt{p}R_{J}\left(0,y,z,p\right)=\frac{3\pi}{2\sqrt{yz}}}}$ \lim_{p\to 0+}\sqrt{p}\CarlsonsymellintRJ@{0}{y}{z}{p} = \frac{3\pi}{2\sqrt{yz}}		Error	Limit[Sqrt[p]*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0], p -> 0, Direction -> "FromAbove", GenerateConditions->None] == Divide[3*Pi,2*Sqrt[y*(x + y*I)]]	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.20#Ex20	${\displaystyle{\displaystyle\lim_{p\to 0-}R_{J}\left(0,y,z,p\right)={-R_{D}\left(0,y,z\right)-R_{D}\left(0,z,y\right)}}}$ \lim_{p\to 0-}\CarlsonsymellintRJ@{0}{y}{z}{p} = {-\CarlsonsymellintRD@{0}{y}{z}-\CarlsonsymellintRD@{0}{z}{y}}		Error	Limit[3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0], p -> 0, Direction -> "FromBelow", GenerateConditions->None] == - 3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))- 3*(EllipticF[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)]-EllipticE[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)])/((y-x + y*I)*(y-0)^(1/2))	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.20#Ex20	${\displaystyle{\displaystyle{-R_{D}\left(0,y,z\right)-R_{D}\left(0,z,y\right)}=\frac{-6}{yz}R_{G}\left(0,y,z\right)}}$ {-\CarlsonsymellintRD@{0}{y}{z}-\CarlsonsymellintRD@{0}{z}{y}} = \frac{-6}{yz}\CarlsonsymellintRG@{0}{y}{z}		Error	- 3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))- 3*(EllipticF[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)]-EllipticE[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)])/((y-x + y*I)*(y-0)^(1/2)) == Divide[- 6,y*(x + y*I)]*Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2])	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Plus[Complex[1.5111033799217843, -0.47027281525563985], Times[Complex[-2.537302274660022, -1.050985014004285], Plus[Complex[1.465481142300126, -0.24396122198922798], Times[Complex[0.2643318009908678, -0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Plus[Complex[-0.13967540286775149, -0.9399293972008751], Times[Complex[2.537302274660022, -1.050985014004285], Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20.E12	${\displaystyle{\displaystyle\lim_{p\to+\infty}pR_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right)}}$ \lim_{p\to+\infty}p\CarlsonsymellintRJ@{x}{y}{z}{p} = 3\CarlsonsymellintRF@{x}{y}{z}		Error	Limit[p*3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x], p -> + Infinity, GenerateConditions->None] == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.20.E12	${\displaystyle{\displaystyle\lim_{p\to-\infty}pR_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right)}}$ \lim_{p\to-\infty}p\CarlsonsymellintRJ@{x}{y}{z}{p} = 3\CarlsonsymellintRF@{x}{y}{z}		Error	Limit[p*3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x], p -> - Infinity, GenerateConditions->None] == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]	Missing Macro Error	Aborted	-	Skipped - Because timed out
19.20.E13	${\displaystyle{\displaystyle 2(p-x)R_{J}\left(x,y,z,p\right)=3R_{F}\left(x,y,z\right)-3\sqrt{x}R_{C}\left(yz,p^{2}\right)}}$ 2(p-x)\CarlsonsymellintRJ@{x}{y}{z}{p} = 3\CarlsonsymellintRF@{x}{y}{z}-3\sqrt{x}\CarlsonellintRC@{yz}{p^{2}}		Error	2*(p - x)*3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]- 3*Sqrt[x]*1/Sqrt[(p)^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(y*(x + y*I))/((p)^(2))]	Missing Macro Error	Aborted	-	Failed [180 / 180] Result: Complex[3.989482635019833, -4.816521080718802] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[5.152296981249878, -0.7434346709776987] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20.E14	${\displaystyle{\displaystyle(q+z)R_{J}\left(x,y,z,-q\right)=(p-z)R_{J}\left(x,y,z,p\right)-3R_{F}\left(x,y,z\right)+3\left(\frac{xyz}{xy+pq}\right)^{1/2}R_{C}\left(xy+pq,pq\right)}}$ (q+z)\CarlsonsymellintRJ@{x}{y}{z}{-q} = (p-z)\CarlsonsymellintRJ@{x}{y}{z}{p}-3\CarlsonsymellintRF@{x}{y}{z}+3\left(\frac{xyz}{xy+pq}\right)^{1/2}\CarlsonellintRC@{xy+pq}{pq}		Error	(q +(x + y*I))*3*(x + y*I-x)/(x + y*I-- q)*(EllipticPi[(x + y*I-- q)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == (p -(x + y*I))*3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]- 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]+ 3*(Divide[x*y*(x + y*I),x*y + p*q])^(1/2)* 1/Sqrt[p*q]*Hypergeometric2F1[1/2,1/2,3/2,1-(x*y + p*q)/(p*q)]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Complex[-3.4116287326863786, 8.252883937385896] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-8.900891250450524, -2.579723477019983] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex21	${\displaystyle{\displaystyle q>0}}$ q > 0		q > 0	q > 0	Skipped - no semantic math	Skipped - no semantic math	-	-
19.20#Ex22	${\displaystyle{\displaystyle p=\frac{z(x+y+q)-xy}{z+q}}}$ p = \frac{z(x+y+q)-xy}{z+q}		p = ((x + y*I)*(x + y + q)- x*y)/((x + y*I)+ q)	p == Divide[(x + y*I)*(x + y + q)- x*y,(x + y*I)+ q]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.20#Ex23	${\displaystyle{\displaystyle p=wy+(1-w)z}}$ p = wy+(1-w)z		p = w*y +(1 - w)*(x + y*I)	p == w*y +(1 - w)*(x + y*I)	Skipped - no semantic math	Skipped - no semantic math	-	-
19.20#Ex24	${\displaystyle{\displaystyle w=\frac{z-x}{z+q}}}$ w = \frac{z-x}{z+q}		w = ((x + y*I)- x)/((x + y*I)+ q)	w == Divide[(x + y*I)- x,(x + y*I)+ q]	Skipped - no semantic math	Skipped - no semantic math	-	-
19.20#Ex25	${\displaystyle{\displaystyle 0<w}}$ 0 < w		0 < w	0 < w	Skipped - no semantic math	Skipped - no semantic math	-	-
19.20.E17	${\displaystyle{\displaystyle(q+z)R_{J}\left(0,y,z,-q\right)=(p-z)R_{J}\left(0,y,z,p\right)-3R_{F}\left(0,y,z\right)}}$ (q+z)\CarlsonsymellintRJ@{0}{y}{z}{-q} = (p-z)\CarlsonsymellintRJ@{0}{y}{z}{p}-3\CarlsonsymellintRF@{0}{y}{z}	${\displaystyle{\displaystyle p=z(y+q)/(z+q),w=z/(z+q)}}$	Error	(q +(x + y*I))*3*(x + y*I-0)/(x + y*I-- q)*(EllipticPi[(x + y*I-- q)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == (p -(x + y*I))*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0]- 3*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[-3.556352843352318, 3.1308549992075583] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-7.694083210877473, -5.44447388199589] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20#Ex26	${\displaystyle{\displaystyle R_{D}\left(x,x,x\right)=x^{-3/2}}}$ \CarlsonsymellintRD@{x}{x}{x} = x^{-3/2}		Error	3*(EllipticF[ArcCos[Sqrt[x/x]],(x-x)/(x-x)]-EllipticE[ArcCos[Sqrt[x/x]],(x-x)/(x-x)])/((x-x)*(x-x)^(1/2)) == (x)^(- 3/2)	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[x, 1.5]} Result: Indeterminate Test Values: {Rule[x, 0.5]} ... skip entries to safe data
19.20#Ex27	${\displaystyle{\displaystyle R_{D}\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{-3/2}R_{D}\left(x,y,z\right)}}$ \CarlsonsymellintRD@{\lambda x}{\lambda y}{\lambda z} = \lambda^{-3/2}\CarlsonsymellintRD@{x}{y}{z}		Error	3*(EllipticF[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)]-EllipticE[ArcCos[Sqrt[\[Lambda]*x/\[Lambda]*(x + y*I)]],(\[Lambda]*(x + y*I)-\[Lambda]*y)/(\[Lambda]*(x + y*I)-\[Lambda]*x)])/((\[Lambda]*(x + y*I)-\[Lambda]*y)*(\[Lambda]*(x + y*I)-\[Lambda]*x)^(1/2)) == \[Lambda]^(- 3/2)* 3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2))	Missing Macro Error	Failure	-	Failed [180 / 180] Result: Complex[1.0149076549010991, -0.8161311339182895] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-1.2947399441897933, -0.14055622592761496] Test Values: {Rule[x, 1.5], Rule[y, -1.5], Rule[λ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20#Ex29	${\displaystyle{\displaystyle R_{D}\left(0,0,z\right)=\infty}}$ \CarlsonsymellintRD@{0}{0}{z} = \infty		Error	3*(EllipticF[ArcCos[Sqrt[0/z]],(z-0)/(z-0)]-EllipticE[ArcCos[Sqrt[0/z]],(z-0)/(z-0)])/((z-0)*(z-0)^(1/2)) == Infinity	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Indeterminate Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.20.E20	${\displaystyle{\displaystyle R_{D}\left(x,y,y\right)=\frac{3}{2(y-x)}\left(R_{C}\left(x,y\right)-\frac{\sqrt{x}}{y}\right)}}$ \CarlsonsymellintRD@{x}{y}{y} = \frac{3}{2(y-x)}\left(\CarlsonellintRC@{x}{y}-\frac{\sqrt{x}}{y}\right)	${\displaystyle{\displaystyle x\neq y,y\neq 0}}$	Error	3*(EllipticF[ArcCos[Sqrt[x/y]],(y-y)/(y-x)]-EllipticE[ArcCos[Sqrt[x/y]],(y-y)/(y-x)])/((y-y)*(y-x)^(1/2)) == Divide[3,2*(y - x)]*(1/Sqrt[y]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y)]-Divide[Sqrt[x],y])	Missing Macro Error	Failure	-	Failed [15 / 15] Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, -0.5]} ... skip entries to safe data
19.20.E21	${\displaystyle{\displaystyle R_{D}\left(x,x,z\right)=\frac{3}{z-x}\left(R_{C}\left(z,x\right)-\frac{1}{\sqrt{z}}\right)}}$ \CarlsonsymellintRD@{x}{x}{z} = \frac{3}{z-x}\left(\CarlsonellintRC@{z}{x}-\frac{1}{\sqrt{z}}\right)	${\displaystyle{\displaystyle x\neq z,xz\neq 0}}$	Error	3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-x)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-x)/(x + y*I-x)])/((x + y*I-x)*(x + y*I-x)^(1/2)) == Divide[3,(x + y*I)- x]*(1/Sqrt[x]*Hypergeometric2F1[1/2,1/2,3/2,1-(x + y*I)/(x)]-Divide[1,Sqrt[x + y*I]])	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[0.13486015646372063, -0.8506635330353051] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[0.13486015646372096, 0.8506635330353054] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.20.E22	${\displaystyle{\displaystyle\int_{0}^{1}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{4}}}=\tfrac{1}{3}R_{D}\left(0,2,1\right)}}$ \int_{0}^{1}\frac{t^{2}\diff{t}}{\sqrt{1-t^{4}}} = \tfrac{1}{3}\CarlsonsymellintRD@{0}{2}{1}		Error	Integrate[Divide[(t)^(2),Sqrt[1 - (t)^(4)]], {t, 0, 1}, GenerateConditions->None] == Divide[1,3]*3*(EllipticF[ArcCos[Sqrt[0/1]],(1-2)/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-2)/(1-0)])/((1-2)*(1-0)^(1/2))	Missing Macro Error	Successful	Skip - symbolical successful subtest	Successful [Tested: 1]
19.20.E22	${\displaystyle{\displaystyle\tfrac{1}{3}R_{D}\left(0,2,1\right)=\frac{\left(\Gamma\left(\frac{3}{4}\right)\right)^{2}}{(2\pi)^{1/2}}}}$ \tfrac{1}{3}\CarlsonsymellintRD@{0}{2}{1} = \frac{\left(\EulerGamma@{\frac{3}{4}}\right)^{2}}{(2\pi)^{1/2}}		Error	Divide[1,3]*3*(EllipticF[ArcCos[Sqrt[0/1]],(1-2)/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-2)/(1-0)])/((1-2)*(1-0)^(1/2)) == Divide[(Gamma[Divide[3,4]])^(2),(2*Pi)^(1/2)]	Missing Macro Error	Successful	Skip - symbolical successful subtest	Successful [Tested: 1]
19.20.E22	${\displaystyle{\displaystyle\frac{\left(\Gamma\left(\frac{3}{4}\right)\right)^{2}}{(2\pi)^{1/2}}=0.59907\;01173\;67796\;10371\dots}}$ \frac{\left(\EulerGamma@{\frac{3}{4}}\right)^{2}}{(2\pi)^{1/2}} = 0.59907\;01173\;67796\;10371\dots		((GAMMA((3)/(4)))^(2))/((2*Pi)^(1/2)) = 0.59907011736779610371	Divide[(Gamma[Divide[3,4]])^(2),(2*Pi)^(1/2)] == 0.59907011736779610371	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]
19.21.E1	${\displaystyle{\displaystyle R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right)+R_{D}\left(0,z+1,z\right)R_{F}\left(0,z+1,1\right)=3\pi/(2z)}}$ \CarlsonsymellintRF@{0}{z+1}{z}\CarlsonsymellintRD@{0}{z+1}{1}+\CarlsonsymellintRD@{0}{z+1}{z}\CarlsonsymellintRF@{0}{z+1}{1} = 3\pi/(2z)		Error	EllipticF[ArcCos[Sqrt[0/z]],(z-z + 1)/(z-0)]/Sqrt[z-0]*3*(EllipticF[ArcCos[Sqrt[0/1]],(1-z + 1)/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-z + 1)/(1-0)])/((1-z + 1)*(1-0)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[0/z]],(z-z + 1)/(z-0)]-EllipticE[ArcCos[Sqrt[0/z]],(z-z + 1)/(z-0)])/((z-z + 1)*(z-0)^(1/2))*EllipticF[ArcCos[Sqrt[0/1]],(1-z + 1)/(1-0)]/Sqrt[1-0] == 3*Pi/(2*z)	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Complex[-18.895019118218656, -13.266297761785948] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-2.405668177707024, 11.584123712813607] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.21.E2	${\displaystyle{\displaystyle 3R_{F}\left(0,y,z\right)=zR_{D}\left(0,y,z\right)+yR_{D}\left(0,z,y\right)}}$ 3\CarlsonsymellintRF@{0}{y}{z} = z\CarlsonsymellintRD@{0}{y}{z}+y\CarlsonsymellintRD@{0}{z}{y}		Error	3*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0] == (x + y*I)*3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))+ y*3*(EllipticF[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)]-EllipticE[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)])/((y-x + y*I)*(y-0)^(1/2))	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[-0.11482200178525792, 3.077769310376559] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[0.9930498831204495, -3.2293137034341144] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E3	${\displaystyle{\displaystyle 6R_{G}\left(0,y,z\right)=yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))}}$ 6\CarlsonsymellintRG@{0}{y}{z} = yz(\CarlsonsymellintRD@{0}{y}{z}+\CarlsonsymellintRD@{0}{z}{y})		Error	6*Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2]) == y*(x + y*I)*(3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)]-EllipticE[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)])/((y-x + y*I)*(y-0)^(1/2)))	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Plus[Complex[-2.3418687704988255, 4.458096439149204], Times[Complex[8.07364639949469, -3.344213836475408], Plus[Complex[1.465481142300126, -0.24396122198922798], Times[Complex[0.2643318009908678, -0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Plus[Complex[1.800571487249528, -2.4291108001544095], Times[Complex[8.07364639949469, 3.344213836475408], Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E3	${\displaystyle{\displaystyle yz(R_{D}\left(0,y,z\right)+R_{D}\left(0,z,y\right))=3zR_{F}\left(0,y,z\right)+z(y-z)R_{D}\left(0,y,z\right)}}$ yz(\CarlsonsymellintRD@{0}{y}{z}+\CarlsonsymellintRD@{0}{z}{y}) = 3z\CarlsonsymellintRF@{0}{y}{z}+z(y-z)\CarlsonsymellintRD@{0}{y}{z}		Error	y*(x + y*I)*(3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)]-EllipticE[ArcCos[Sqrt[0/y]],(y-x + y*I)/(y-0)])/((y-x + y*I)*(y-0)^(1/2))) == 3*(x + y*I)*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]+(x + y*I)*(y -(x + y*I))*3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[-4.444420962886951, -4.788886968242726] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-6.333545379831845, 3.3543957304704977] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E4	${\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)-\mathrm{i}R_{F}\left(0,z,1\right)}}$ \CarlsonsymellintRF@{0}{z-1}{z} = \CarlsonsymellintRF@{0}{1-z}{1}-\iunit\CarlsonsymellintRF@{0}{z}{1}		0.5*int(1/(sqrt(t+0)*sqrt(t+z - 1)*sqrt(t+z)), t = 0..infinity) = 0.5*int(1/(sqrt(t+0)*sqrt(t+1 - z)*sqrt(t+1)), t = 0..infinity)- I*0.5*int(1/(sqrt(t+0)*sqrt(t+z)*sqrt(t+1)), t = 0..infinity)	EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]/Sqrt[z-0] == EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]- I*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]	Failure	Failure	Failed [7 / 7] Result: 3.197606220-2.012137137*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: 1.024722154-2.538160454*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.447882135735306, 1.6422203572966838] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.9112982419758283, 0.8007121244739206] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.21.E4	${\displaystyle{\displaystyle R_{F}\left(0,z-1,z\right)=R_{F}\left(0,1-z,1\right)+\mathrm{i}R_{F}\left(0,z,1\right)}}$ \CarlsonsymellintRF@{0}{z-1}{z} = \CarlsonsymellintRF@{0}{1-z}{1}+\iunit\CarlsonsymellintRF@{0}{z}{1}		0.5*int(1/(sqrt(t+0)*sqrt(t+z - 1)*sqrt(t+z)), t = 0..infinity) = 0.5*int(1/(sqrt(t+0)*sqrt(t+1 - z)*sqrt(t+1)), t = 0..infinity)+ I*0.5*int(1/(sqrt(t+0)*sqrt(t+z)*sqrt(t+1)), t = 0..infinity)	EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]/Sqrt[z-0] == EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]+ I*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]	Failure	Failure	Failed [7 / 7] Result: 3.609429842+1.115973839*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: 2.710472508+.381644808*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.0036174998504115152, -2.054982714938571] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.9086726238549093, -2.7316000638683375] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.21.E5	${\displaystyle{\displaystyle 2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1\right)+\mathrm{i}2R_{G}\left(0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)-\mathrm{i}zR_{F}\left(0,z,1\right)}}$ 2\CarlsonsymellintRG@{0}{z-1}{z} = 2\CarlsonsymellintRG@{0}{1-z}{1}+\iunit 2\CarlsonsymellintRG@{0}{z}{1}+(z-1)\CarlsonsymellintRF@{0}{1-z}{1}-\iunit z\CarlsonsymellintRF@{0}{z}{1}		Error	2*Sqrt[z-0]*(EllipticE[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+(Cot[ArcCos[Sqrt[0/z]]])^2*EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+Cot[ArcCos[Sqrt[0/z]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/z]]]^2]) == 2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])+ I*2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])+(z - 1)*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]- I*z*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Complex[0.23313173408598564, -1.9381268036446178] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.6842698833888152, -2.1985132995849304] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.21.E5	${\displaystyle{\displaystyle 2R_{G}\left(0,z-1,z\right)=2R_{G}\left(0,1-z,1\right)-\mathrm{i}2R_{G}\left(0,z,1\right)+(z-1)R_{F}\left(0,1-z,1\right)+\mathrm{i}zR_{F}\left(0,z,1\right)}}$ 2\CarlsonsymellintRG@{0}{z-1}{z} = 2\CarlsonsymellintRG@{0}{1-z}{1}-\iunit 2\CarlsonsymellintRG@{0}{z}{1}+(z-1)\CarlsonsymellintRF@{0}{1-z}{1}+\iunit z\CarlsonsymellintRF@{0}{z}{1}		Error	2*Sqrt[z-0]*(EllipticE[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+(Cot[ArcCos[Sqrt[0/z]]])^2*EllipticF[ArcCos[Sqrt[0/z]],(z-z - 1)/(z-0)]+Cot[ArcCos[Sqrt[0/z]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/z]]]^2]) == 2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])- I*2*Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2])+(z - 1)*EllipticF[ArcCos[Sqrt[0/1]],(1-1 - z)/(1-0)]/Sqrt[1-0]+ I*z*EllipticF[ArcCos[Sqrt[0/1]],(1-z)/(1-0)]/Sqrt[1-0]	Missing Macro Error	Failure	-	Failed [7 / 7] Result: Complex[0.44709928924442033, 1.6621495887309192] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.1273665829731985, 1.9939163092606038] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.21.E6	${\displaystyle{\displaystyle(\sqrt{rp}/z)R_{J}\left(0,y,z,p\right)={(r-1)}R_{F}\left(0,y,z\right)R_{D}\left(p,rz,z\right)+R_{D}\left(0,y,z\right)R_{F}\left(p,rz,z\right)}}$ (\sqrt{rp}/z)\CarlsonsymellintRJ@{0}{y}{z}{p} = {(r-1)}\CarlsonsymellintRF@{0}{y}{z}\CarlsonsymellintRD@{p}{rz}{z}+\CarlsonsymellintRD@{0}{y}{z}\CarlsonsymellintRF@{p}{rz}{z}	${\displaystyle{\displaystyle r=(y-p)/(y-z),(y-p)/(y-z)>0}}$	Error	(Sqrt[r*p]/(x + y*I))*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == (r - 1)*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]*3*(EllipticF[ArcCos[Sqrt[p/x + y*I]],(x + y*I-r*(x + y*I))/(x + y*I-p)]-EllipticE[ArcCos[Sqrt[p/x + y*I]],(x + y*I-r*(x + y*I))/(x + y*I-p)])/((x + y*I-r*(x + y*I))*(x + y*I-p)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2))*EllipticF[ArcCos[Sqrt[p/x + y*I]],(x + y*I-r*(x + y*I))/(x + y*I-p)]/Sqrt[x + y*I-p]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Complex[-0.019107479205769995, -0.26821779662698253] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[r, -1.5], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[1.626010920193221, 0.604709928225457] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[r, -1.5], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E7	${\displaystyle{\displaystyle(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z\right)-3\sqrt{y/(xz)}}}$ (x-y)\CarlsonsymellintRD@{y}{z}{x}+(z-y)\CarlsonsymellintRD@{x}{y}{z} = 3\CarlsonsymellintRF@{x}{y}{z}-3\sqrt{y/(xz)}		Error	(x - y)*3*(EllipticF[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)]-EllipticE[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)])/((x-x + y*I)*(x-y)^(1/2))+((x + y*I)- y)*3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2)) == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]- 3*Sqrt[y/(x*(x + y*I))]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[2.01816993619941, -7.647648832317454] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[1.9029767059950156, 2.211761239496786] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E8	${\displaystyle{\displaystyle R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3(xyz)^{-1/2}}}$ \CarlsonsymellintRD@{y}{z}{x}+\CarlsonsymellintRD@{z}{x}{y}+\CarlsonsymellintRD@{x}{y}{z} = 3(xyz)^{-1/2}		Error	3*(EllipticF[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)]-EllipticE[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)])/((x-x + y*I)*(x-y)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)]-EllipticE[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)])/((y-x)*(y-x + y*I)^(1/2))+ 3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2)) == 3*(x*y*(x + y*I))^(- 1/2)	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Complex[0.061772053426947915, 0.22732915812456822] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E9	${\displaystyle{\displaystyle xR_{D}\left(y,z,x\right)+yR_{D}\left(z,x,y\right)+zR_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z\right)}}$ x\CarlsonsymellintRD@{y}{z}{x}+y\CarlsonsymellintRD@{z}{x}{y}+z\CarlsonsymellintRD@{x}{y}{z} = 3\CarlsonsymellintRF@{x}{y}{z}		Error	x*3*(EllipticF[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)]-EllipticE[ArcCos[Sqrt[y/x]],(x-x + y*I)/(x-y)])/((x-x + y*I)*(x-y)^(1/2))+ y*3*(EllipticF[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)]-EllipticE[ArcCos[Sqrt[x + y*I/y]],(y-x)/(y-x + y*I)])/((y-x)*(y-x + y*I)^(1/2))+(x + y*I)*3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2)) == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]	Missing Macro Error	Aborted	-	Failed [18 / 18] Result: Complex[0.3490343350525606, -4.182689157514275] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Indeterminate Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E10	${\displaystyle{\displaystyle 2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)R_{D}\left(x,y,z\right)+\sqrt{xy/z}}}$ 2\CarlsonsymellintRG@{x}{y}{z} = z\CarlsonsymellintRF@{x}{y}{z}-\tfrac{1}{3}(x-z)(y-z)\CarlsonsymellintRD@{x}{y}{z}+\sqrt{xy/z}	${\displaystyle{\displaystyle z\neq 0}}$	Error	2*Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) == (x + y*I)*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]-Divide[1,3]*(x -(x + y*I))*(y -(x + y*I))*3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2))+Sqrt[x*y/(x + y*I)]	Missing Macro Error	Failure	-	Failed [18 / 18] Result: Plus[Complex[-2.045465659795318, -0.2973389532409781], Times[Complex[1.7320508075688772, -1.732050807568877], Plus[Complex[0.9985512968581824, 0.2012315241723115], Times[Complex[0.3176872874027722, -1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, -1.5]} Result: Plus[Complex[-2.0191372830755783, 1.5655011975568338], Times[Complex[1.7320508075688772, 1.732050807568877], Plus[Complex[1.0566228789425183, 0.3443432776585209], Times[Complex[0.3176872874027722, 1.049249833251038], Power[Plus[1.0, Times[Complex[0.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]] Test Values: {Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data
19.21.E12	${\displaystyle{\displaystyle(p-x)R_{J}\left(x,y,z,p\right)+(q-x)R_{J}\left(x,y,z,q\right)=3R_{F}\left(x,y,z\right)-3R_{C}\left(\xi,\eta\right)}}$ (p-x)\CarlsonsymellintRJ@{x}{y}{z}{p}+(q-x)\CarlsonsymellintRJ@{x}{y}{z}{q} = 3\CarlsonsymellintRF@{x}{y}{z}-3\CarlsonellintRC@{\xi}{\eta}		Error	(p - x)*3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]+(q - x)*3*(x + y*I-x)/(x + y*I-q)*(EllipticPi[(x + y*I-q)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x] == 3*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]- 3*1/Sqrt[\[Eta]]*Hypergeometric2F1[1/2,1/2,3/2,1-(\[Xi])/(\[Eta])]	Missing Macro Error	Aborted	-	Failed [300 / 300] Result: Indeterminate Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[η, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ξ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[5.153237655786464, -3.718995107844719] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5], Rule[η, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ξ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
19.21#Ex1	${\displaystyle{\displaystyle(p-x)(q-x)=(y-x)(z-x)}}$ (p-x)(q-x) = (y-x)(z-x)		(p - x)*(q - x) = (y - x)*((x + y*I)- x)	(p - x)*(q - x) == (y - x)*((x + y*I)- x)	Skipped - no semantic math	Skipped - no semantic math	-	-
19.21#Ex2	${\displaystyle{\displaystyle\xi=yz/x}}$ \xi = yz/x		xi = y*z/x	\[Xi] == y*z/x	Skipped - no semantic math	Skipped - no semantic math	-	-
19.21#Ex3	${\displaystyle{\displaystyle\eta=pq/x}}$ \eta = pq/x		eta = p*q/x	\[Eta] == p*q/x	Skipped - no semantic math	Skipped - no semantic math	-	-
19.21.E14	${\displaystyle{\displaystyle\eta-\xi=p+q-y-z}}$ \eta-\xi = p+q-y-z		eta - xi = p + q - y -(x + y*I)	\[Eta]- \[Xi] == p + q - y -(x + y*I)	Skipped - no semantic math	Skipped - no semantic math	-	-
19.21.E15	${\displaystyle{\displaystyle pR_{J}\left(0,y,z,p\right)+qR_{J}\left(0,y,z,q\right)=3R_{F}\left(0,y,z\right)}}$ p\CarlsonsymellintRJ@{0}{y}{z}{p}+q\CarlsonsymellintRJ@{0}{y}{z}{q} = 3\CarlsonsymellintRF@{0}{y}{z}	${\displaystyle{\displaystyle pq=yz}}$	Error	p*3*(x + y*I-0)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0]+ q*3*(x + y*I-0)/(x + y*I-q)*(EllipticPi[(x + y*I-q)/(x + y*I-0),ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/Sqrt[x + y*I-0] == 3*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]	Missing Macro Error	Failure	-	Failed [300 / 300] Result: Complex[-0.5878632565330948, -3.2355968294614907] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[-2.320767562800481, 3.5603464097743847] Test Values: {Rule[p, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[q, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[x, 1.5], Rule[y, 1.5]} ... skip entries to safe data