19.2

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