Results of Elementary Functions II

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.24.E2	${\displaystyle{\displaystyle\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right)}}$ \acos@@{z} = (2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right)	${\displaystyle{\displaystyle|1-z|\leq 2}}$	arccos(z) = (2*(1 - z))^(1/2)*(1 + sum((1 * 3 * 5*(2*n - 1))/((2)^(2*n)*(2*n + 1)*factorial(n))*(1 - z)^(n), n = 1..infinity))	ArcCos[z] == (2*(1 - z))^(1/2)*(1 + Sum[Divide[1 * 3 * 5*(2*n - 1),(2)^(2*n)*(2*n + 1)*(n)!]*(1 - z)^(n), {n, 1, Infinity}, GenerateConditions->None])	Failure	Failure	Failed [7 / 7] Result: .3065228369+.5552108774*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -3.012742443+4.300365362*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: .3498215011-1.819822265*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: -5.876013992-3.037981862*I Test Values: {z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.3065228364484756, 0.5552108781095243] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-3.0127424460165777, 4.300365361528893] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.24.E6	${\displaystyle{\displaystyle x^{2}-y^{2}=-\tfrac{1}{2}}}$ x^{2}-y^{2} = -\tfrac{1}{2}		(x)^(2)- (y)^(2) = -(1)/(2)	(x)^(2)- (y)^(2) == -Divide[1,2]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.24.E7	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsin}z=(1-z^{2})^{-1/2}}}$ \deriv{}{z}\asin@@{z} = (1-z^{2})^{-1/2}		diff(arcsin(z), z) = (1 - (z)^(2))^(- 1/2)	D[ArcSin[z], z] == (1 - (z)^(2))^(- 1/2)	Successful	Successful	-	Successful [Tested: 7]
4.24.E8	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccos}z=-(1-z^{2})^{-1/2}}}$ \deriv{}{z}\acos@@{z} = -(1-z^{2})^{-1/2}		diff(arccos(z), z) = -(1 - (z)^(2))^(- 1/2)	D[ArcCos[z], z] == -(1 - (z)^(2))^(- 1/2)	Successful	Successful	-	Successful [Tested: 7]
4.24.E9	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctan}z=\frac{1}{1+z^{2}}}}$ \deriv{}{z}\atan@@{z} = \frac{1}{1+z^{2}}		diff(arctan(z), z) = (1)/(1 + (z)^(2))	D[ArcTan[z], z] == Divide[1,1 + (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.24.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc}z=-\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\acsc@@{z} = -\frac{1}{z(z^{2}-1)^{1/2}}		diff(arccsc(z), z) = -(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcCsc[z], z] == -Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [2 / 7] Result: 1.074569932-1.074569932*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: -.6696152420e-9+2.000000000*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.24.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc}z=+\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\acsc@@{z} = +\frac{1}{z(z^{2}-1)^{1/2}}		diff(arccsc(z), z) = +(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcCsc[z], z] == +Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [5 / 7] Result: -.6696152420e-9+2.000000000*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: 1.074569932-1.074569932*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: -1.192569588 Test Values: {z = 1.5, z = 1/2} Result: 4.618802153*I Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: Complex[0.0, 4.618802153517007] Test Values: {Rule[z, Rational[1, 2]]}
4.24.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec}z=+\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\asec@@{z} = +\frac{1}{z(z^{2}-1)^{1/2}}		diff(arcsec(z), z) = +(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcSec[z], z] == +Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [2 / 7] Result: -1.074569932+1.074569932*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: .6696152420e-9-2.000000000*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.24.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec}z=-\frac{1}{z(z^{2}-1)^{1/2}}}}$ \deriv{}{z}\asec@@{z} = -\frac{1}{z(z^{2}-1)^{1/2}}		diff(arcsec(z), z) = -(1)/(z*((z)^(2)- 1)^(1/2))	D[ArcSec[z], z] == -Divide[1,z*((z)^(2)- 1)^(1/2)]	Failure	Failure	Failed [5 / 7] Result: .6696152420e-9-2.000000000*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: -1.074569932+1.074569932*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: 1.192569588 Test Values: {z = 1.5, z = 1/2} Result: -4.618802153*I Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: Complex[0.0, -4.618802153517007] Test Values: {Rule[z, Rational[1, 2]]}
4.24.E12	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccot}z=-\frac{1}{1+z^{2}}}}$ \deriv{}{z}\acot@@{z} = -\frac{1}{1+z^{2}}		diff(arccot(z), z) = -(1)/(1 + (z)^(2))	D[ArcCot[z], z] == -Divide[1,1 + (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.24.E13	${\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1-v^{2})^{1/2}+v(1-u^{2})^{1/2}\right)}}$ \Asin@@{u}+\Asin@@{v} = \Asin@{u(1-v^{2})^{1/2}+ v(1-u^{2})^{1/2}}		Error	ArcSin[u]+ ArcSin[v] == ArcSin[u*(1 - (v)^(2))^(1/2)+ v*(1 - (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [34 / 100] Result: Complex[4.440892098500626*^-16, 2.633915793849633] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.5707963267948966, -0.6078894033135972] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, 1.5]} ... skip entries to safe data
4.24.E13	${\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1-v^{2})^{1/2}-v(1-u^{2})^{1/2}\right)}}$ \Asin@@{u}-\Asin@@{v} = \Asin@{u(1-v^{2})^{1/2}- v(1-u^{2})^{1/2}}		Error	ArcSin[u]- ArcSin[v] == ArcSin[u*(1 - (v)^(2))^(1/2)- v*(1 - (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [34 / 100] Result: Complex[4.440892098500626*^-16, 2.633915793849633] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: Complex[1.5707963267948966, -0.6078894033135972] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]} ... skip entries to safe data
4.24.E14	${\displaystyle{\displaystyle\operatorname{Arccos}u+\operatorname{Arccos}v=\operatorname{Arccos}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Acos@@{u}+\Acos@@{v} = \Acos@{uv-((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcCos[u]+ ArcCos[v] == ArcCos[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[1.5707963267948963, -3.2418051971632305] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]} Result: Complex[1.5707963267948966, -3.9508736907744497] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -2]} ... skip entries to safe data
4.24.E14	${\displaystyle{\displaystyle\operatorname{Arccos}u-\operatorname{Arccos}v=\operatorname{Arccos}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Acos@@{u}-\Acos@@{v} = \Acos@{uv+((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcCos[u]- ArcCos[v] == ArcCos[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[-2.3202651922123767, 0.3459279941338042] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8213274613774166, -2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E15	${\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arctan}v=\operatorname{Arctan}\left(\frac{u+v}{1-uv}\right)}}$ \Atan@@{u}+\Atan@@{v} = \Atan@{\frac{u+ v}{1- uv}}		Error	ArcTan[u]+ ArcTan[v] == ArcTan[Divide[u + v,1 - u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.24.E15	${\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arctan}v=\operatorname{Arctan}\left(\frac{u-v}{1+uv}\right)}}$ \Atan@@{u}-\Atan@@{v} = \Atan@{\frac{u- v}{1+ uv}}		Error	ArcTan[u]- ArcTan[v] == ArcTan[Divide[u - v,1 + u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Asin@@{u}+\Acos@@{v} = \Asin@{uv+((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcSin[u]+ ArcCos[v] == ArcSin[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[2.3202651922123767, -0.3459279941338042] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.8213274613774166, 2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}}$ \Asin@@{u}-\Acos@@{v} = \Asin@{uv-((1-u^{2})(1-v^{2}))^{1/2}}		Error	ArcSin[u]- ArcCos[v] == ArcSin[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]	Missing Macro Error	Failure	-	Failed [63 / 100] Result: Complex[-1.5707963267948963, 3.2418051971632305] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]} Result: Complex[-1.5707963267948966, 3.9508736907744497] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -2]} ... skip entries to safe data
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}-u(1-v^{2})^{1/2}\right)}}$ \Asin@{uv+((1-u^{2})(1-v^{2}))^{1/2}} = \Acos@{v(1-u^{2})^{1/2}- u(1-v^{2})^{1/2}}		Error	ArcSin[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)] == ArcCos[v*(1 - (u)^(2))^(1/2)- u*(1 - (v)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [72 / 100] Result: Complex[-2.3202651922123763, 0.3459279941338042] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8213274613774164, -2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E16	${\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}+u(1-v^{2})^{1/2}\right)}}$ \Asin@{uv-((1-u^{2})(1-v^{2}))^{1/2}} = \Acos@{v(1-u^{2})^{1/2}+ u(1-v^{2})^{1/2}}		Error	ArcSin[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)] == ArcCos[v*(1 - (u)^(2))^(1/2)+ u*(1 - (v)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [91 / 100] Result: Complex[-2.3202651922123767, 2.979843787983438] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8213274613774161, -0.3459279941338048] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arccot}v=\operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)}}$ \Atan@@{u}+\Acot@@{v} = \Atan@{\frac{uv+ 1}{v- u}}		Error	ArcTan[u]+ ArcCot[v] == ArcTan[Divide[u*v + 1,v - u]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arccot}v=\operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)}}$ \Atan@@{u}-\Acot@@{v} = \Atan@{\frac{uv- 1}{v+ u}}		Error	ArcTan[u]- ArcCot[v] == ArcTan[Divide[u*v - 1,v + u]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)=\operatorname{Arccot}\left(\frac{v-u}{uv+1}\right)}}$ \Atan@{\frac{uv+ 1}{v- u}} = \Acot@{\frac{v- u}{uv+ 1}}		Error	ArcTan[Divide[u*v + 1,v - u]] == ArcCot[Divide[v - u,u*v + 1]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}
4.24.E17	${\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)=\operatorname{Arccot}\left(\frac{v+u}{uv-1}\right)}}$ \Atan@{\frac{uv- 1}{v+ u}} = \Acot@{\frac{v+ u}{uv- 1}}		Error	ArcTan[Divide[u*v - 1,v + u]] == ArcCot[Divide[v + u,u*v - 1]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}
4.26.E1	${\displaystyle{\displaystyle\int\sin x\mathrm{d}x=-\cos x}}$ \int\sin@@{x}\diff{x} = -\cos@@{x}		int(sin(x), x) = - cos(x)	Integrate[Sin[x], x, GenerateConditions->None] == - Cos[x]	Successful	Successful	-	Successful [Tested: 3]
4.26.E2	${\displaystyle{\displaystyle\int\cos x\mathrm{d}x=\sin x}}$ \int\cos@@{x}\diff{x} = \sin@@{x}		int(cos(x), x) = sin(x)	Integrate[Cos[x], x, GenerateConditions->None] == Sin[x]	Successful	Successful	-	Successful [Tested: 3]
4.26.E3	${\displaystyle{\displaystyle\int\tan x\mathrm{d}x=-\ln\left(\cos x\right)}}$ \int\tan@@{x}\diff{x} = -\ln@{\cos@@{x}}	${\displaystyle{\displaystyle-\tfrac{1}{2}\pi<x,x<\tfrac{1}{2}\pi}}$	int(tan(x), x) = - ln(cos(x))	Integrate[Tan[x], x, GenerateConditions->None] == - Log[Cos[x]]	Successful	Successful	-	Successful [Tested: 2]
4.26.E4	${\displaystyle{\displaystyle\int\csc x\mathrm{d}x=\ln\left(\tan\tfrac{1}{2}x\right)}}$ \int\csc@@{x}\diff{x} = \ln@{\tan@@{\tfrac{1}{2}x}}	${\displaystyle{\displaystyle 0<x,x<\pi}}$	int(csc(x), x) = ln(tan((1)/(2)*x))	Integrate[Csc[x], x, GenerateConditions->None] == Log[Tan[Divide[1,2]*x]]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.26.E5	${\displaystyle{\displaystyle\int\sec x\mathrm{d}x={\operatorname{gd}^{-1}}\left(x\right)}}$ \int\sec@@{x}\diff{x} = \aGudermannian@{x}	${\displaystyle{\displaystyle-\frac{1}{2}\pi<x,x<\frac{1}{2}\pi}}$	int(sec(x), x) = arctanh(sin(x))	Integrate[Sec[x], x, GenerateConditions->None] == InverseGudermannian[x]	Failure	Failure	Successful [Tested: 2]	Successful [Tested: 2]
4.26.E6	${\displaystyle{\displaystyle\int\cot x\mathrm{d}x=\ln\left(\sin x\right)}}$ \int\cot@@{x}\diff{x} = \ln@{\sin@@{x}}	${\displaystyle{\displaystyle 0<x,x<\pi}}$	int(cot(x), x) = ln(sin(x))	Integrate[Cot[x], x, GenerateConditions->None] == Log[Sin[x]]	Successful	Successful	-	Successful [Tested: 3]
4.26.E7	${\displaystyle{\displaystyle\int e^{ax}\sin\left(bx\right)\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\sin\left(bx\right)-b\cos\left(bx\right))}}$ \int e^{ax}\sin@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\sin@{bx}-b\cos@{bx})		int(exp(a*x)*sin(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*sin(b*x)- b*cos(b*x))	Integrate[Exp[a*x]*Sin[b*x], x, GenerateConditions->None] == Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Sin[b*x]- b*Cos[b*x])	Successful	Successful	-	Successful [Tested: 108]
4.26.E8	${\displaystyle{\displaystyle\int e^{ax}\cos\left(bx\right)\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\cos\left(bx\right)+b\sin\left(bx\right))}}$ \int e^{ax}\cos@{bx}\diff{x} = \frac{e^{ax}}{a^{2}+b^{2}}(a\cos@{bx}+b\sin@{bx})		int(exp(a*x)*cos(b*x), x) = (exp(a*x))/((a)^(2)+ (b)^(2))*(a*cos(b*x)+ b*sin(b*x))	Integrate[Exp[a*x]*Cos[b*x], x, GenerateConditions->None] == Divide[Exp[a*x],(a)^(2)+ (b)^(2)]*(a*Cos[b*x]+ b*Sin[b*x])	Successful	Successful	-	Successful [Tested: 108]
4.26.E9	${\displaystyle{\displaystyle\int_{0}^{\pi}\sin\left(mt\right)\sin\left(nt\right)\mathrm{d}t=0}}$ \int_{0}^{\pi}\sin@{mt}\sin@{nt}\diff{t} = 0	${\displaystyle{\displaystyle m\neq n}}$	int(sin(m*t)*sin(n*t), t = 0..Pi) = 0	Integrate[Sin[m*t]*Sin[n*t], {t, 0, Pi}, GenerateConditions->None] == 0	Successful	Failure	-	Successful [Tested: 6]
4.26.E10	${\displaystyle{\displaystyle\int_{0}^{\pi}\cos\left(mt\right)\cos\left(nt\right)\mathrm{d}t=0}}$ \int_{0}^{\pi}\cos@{mt}\cos@{nt}\diff{t} = 0	${\displaystyle{\displaystyle m\neq n}}$	int(cos(m*t)*cos(n*t), t = 0..Pi) = 0	Integrate[Cos[m*t]*Cos[n*t], {t, 0, Pi}, GenerateConditions->None] == 0	Successful	Failure	-	Successful [Tested: 6]
4.26.E11	${\displaystyle{\displaystyle\int_{0}^{\pi}{\sin^{2}}\left(nt\right)\mathrm{d}t=\int_{0}^{\pi}{\cos^{2}}\left(nt\right)\mathrm{d}t}}$ \int_{0}^{\pi}\sin^{2}@{nt}\diff{t} = \int_{0}^{\pi}\cos^{2}@{nt}\diff{t}	${\displaystyle{\displaystyle n\neq 0}}$	int((sin(n*t))^(2), t = 0..Pi) = int((cos(n*t))^(2), t = 0..Pi)	Integrate[(Sin[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None]	Successful	Failure	-	Successful [Tested: 3]
4.26.E11	${\displaystyle{\displaystyle\int_{0}^{\pi}{\cos^{2}}\left(nt\right)\mathrm{d}t=\tfrac{1}{2}\pi}}$ \int_{0}^{\pi}\cos^{2}@{nt}\diff{t} = \tfrac{1}{2}\pi	${\displaystyle{\displaystyle n\neq 0}}$	int((cos(n*t))^(2), t = 0..Pi) = (1)/(2)*Pi	Integrate[(Cos[n*t])^(2), {t, 0, Pi}, GenerateConditions->None] == Divide[1,2]*Pi	Successful	Failure	-	Successful [Tested: 3]
4.26.E13	${\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(t^{2}\right)\mathrm{d}t=\int_{0}^{\infty}\cos\left(t^{2}\right)\mathrm{d}t}}$ \int_{0}^{\infty}\sin@{t^{2}}\diff{t} = \int_{0}^{\infty}\cos@{t^{2}}\diff{t}		int(sin((t)^(2)), t = 0..infinity) = int(cos((t)^(2)), t = 0..infinity)	Integrate[Sin[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 1]
4.26.E13	${\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(t^{2}\right)\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{2}}}}$ \int_{0}^{\infty}\cos@{t^{2}}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{2}}		int(cos((t)^(2)), t = 0..infinity) = (1)/(2)*sqrt((Pi)/(2))	Integrate[Cos[(t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Sqrt[Divide[Pi,2]]	Successful	Successful	-	Successful [Tested: 1]
4.26.E14	${\displaystyle{\displaystyle\int\operatorname{arcsin}x\mathrm{d}x=x\operatorname{arcsin}x+(1-x^{2})^{1/2}}}$ \int\asin@@{x}\diff{x} = x\asin@@{x}+(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(arcsin(x), x) = x*arcsin(x)+(1 - (x)^(2))^(1/2)	Integrate[ArcSin[x], x, GenerateConditions->None] == x*ArcSin[x]+(1 - (x)^(2))^(1/2)	Successful	Successful	-	Successful [Tested: 1]
4.26.E15	${\displaystyle{\displaystyle\int\operatorname{arccos}x\mathrm{d}x=x\operatorname{arccos}x-(1-x^{2})^{1/2}}}$ \int\acos@@{x}\diff{x} = x\acos@@{x}-(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(arccos(x), x) = x*arccos(x)-(1 - (x)^(2))^(1/2)	Integrate[ArcCos[x], x, GenerateConditions->None] == x*ArcCos[x]-(1 - (x)^(2))^(1/2)	Successful	Successful	-	Successful [Tested: 1]
4.26.E16	${\displaystyle{\displaystyle\int\operatorname{arctan}x\mathrm{d}x=x\operatorname{arctan}x-\tfrac{1}{2}\ln\left(1+x^{2}\right)}}$ \int\atan@@{x}\diff{x} = x\atan@@{x}-\tfrac{1}{2}\ln@{1+x^{2}}	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	int(arctan(x), x) = x*arctan(x)-(1)/(2)*ln(1 + (x)^(2))	Integrate[ArcTan[x], x, GenerateConditions->None] == x*ArcTan[x]-Divide[1,2]*Log[1 + (x)^(2)]	Successful	Successful	-	Successful [Tested: 3]
4.26.E17	${\displaystyle{\displaystyle\int\operatorname{arccsc}x\mathrm{d}x=x\operatorname{arccsc}x+\ln\left(x+(x^{2}-1)^{1/2}\right)}}$ \int\acsc@@{x}\diff{x} = x\acsc@@{x}+\ln@{x+(x^{2}-1)^{1/2}}	${\displaystyle{\displaystyle 1<x,x<\infty}}$	int(arccsc(x), x) = x*arccsc(x)+ ln(x +((x)^(2)- 1)^(1/2))	Integrate[ArcCsc[x], x, GenerateConditions->None] == x*ArcCsc[x]+ Log[x +((x)^(2)- 1)^(1/2)]	Successful	Failure	-	Failed [2 / 2] Result: Complex[-1.1102230246251565*^-16, -1.5707963267948966] Test Values: {Rule[x, 1.5]} Result: Complex[-4.440892098500626*^-16, -1.5707963267948966] Test Values: {Rule[x, 2]}
4.26.E18	${\displaystyle{\displaystyle\int\operatorname{arcsec}x\mathrm{d}x=x\operatorname{arcsec}x-\ln\left(x+(x^{2}-1)^{1/2}\right)}}$ \int\asec@@{x}\diff{x} = x\asec@@{x}-\ln@{x+(x^{2}-1)^{1/2}}	${\displaystyle{\displaystyle 1<x,x<\infty}}$	int(arcsec(x), x) = x*arcsec(x)- ln(x +((x)^(2)- 1)^(1/2))	Integrate[ArcSec[x], x, GenerateConditions->None] == x*ArcSec[x]- Log[x +((x)^(2)- 1)^(1/2)]	Successful	Failure	-	Failed [2 / 2] Result: Complex[1.1102230246251565*^-16, 1.5707963267948966] Test Values: {Rule[x, 1.5]} Result: Complex[4.440892098500626*^-16, 1.5707963267948966] Test Values: {Rule[x, 2]}
4.26.E19	${\displaystyle{\displaystyle\int\operatorname{arccot}x\mathrm{d}x=x\operatorname{arccot}x+\tfrac{1}{2}\ln\left(1+x^{2}\right)}}$ \int\acot@@{x}\diff{x} = x\acot@@{x}+\tfrac{1}{2}\ln@{1+x^{2}}	${\displaystyle{\displaystyle 0<x,x<\infty}}$	int(arccot(x), x) = x*arccot(x)+(1)/(2)*ln(1 + (x)^(2))	Integrate[ArcCot[x], x, GenerateConditions->None] == x*ArcCot[x]+Divide[1,2]*Log[1 + (x)^(2)]	Successful	Successful	-	Successful [Tested: 3]
4.26.E20	${\displaystyle{\displaystyle\int x\operatorname{arcsin}x\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{1}{4}\right)\operatorname{arcsin}x+\frac{x}{4}(1-x^{2})^{1/2}}}$ \int x\asin@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\asin@@{x}+\frac{x}{4}(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(x*arcsin(x), x) = (((x)^(2))/(2)-(1)/(4))*arcsin(x)+(x)/(4)*(1 - (x)^(2))^(1/2)	Integrate[x*ArcSin[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcSin[x]+Divide[x,4]*(1 - (x)^(2))^(1/2)	Successful	Successful	-	Successful [Tested: 1]
4.26.E21	${\displaystyle{\displaystyle\int x\operatorname{arccos}x\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{1}{4}\right)\operatorname{arccos}x-\frac{x}{4}(1-x^{2})^{1/2}}}$ \int x\acos@@{x}\diff{x} = \left(\frac{x^{2}}{2}-\frac{1}{4}\right)\acos@@{x}-\frac{x}{4}(1-x^{2})^{1/2}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(x*arccos(x), x) = (((x)^(2))/(2)-(1)/(4))*arccos(x)-(x)/(4)*(1 - (x)^(2))^(1/2)	Integrate[x*ArcCos[x], x, GenerateConditions->None] == (Divide[(x)^(2),2]-Divide[1,4])*ArcCos[x]-Divide[x,4]*(1 - (x)^(2))^(1/2)	Failure	Failure	Failed [1 / 1] Result: .3926990817 Test Values: {x = .5}	Failed [1 / 1] Result: 0.3926990816987242 Test Values: {Rule[x, 0.5]}
4.28.E1	${\displaystyle{\displaystyle\sinh z=\frac{e^{z}-e^{-z}}{2}}}$ \sinh@@{z} = \frac{e^{z}-e^{-z}}{2}		sinh(z) = (exp(z)- exp(- z))/(2)	Sinh[z] == Divide[Exp[z]- Exp[- z],2]	Successful	Successful	-	Successful [Tested: 7]
4.28.E2	${\displaystyle{\displaystyle\cosh z=\frac{e^{z}+e^{-z}}{2}}}$ \cosh@@{z} = \frac{e^{z}+e^{-z}}{2}		cosh(z) = (exp(z)+ exp(- z))/(2)	Cosh[z] == Divide[Exp[z]+ Exp[- z],2]	Successful	Successful	-	Successful [Tested: 7]
4.28.E3	${\displaystyle{\displaystyle\cosh z+\sinh z=e^{+z}}}$ \cosh@@{z}+\sinh@@{z} = e^{+ z}		cosh(z)+ sinh(z) = exp(+ z)	Cosh[z]+ Sinh[z] == Exp[+ z]	Successful	Successful	-	Successful [Tested: 7]
4.28.E3	${\displaystyle{\displaystyle\cosh z-\sinh z=e^{-z}}}$ \cosh@@{z}-\sinh@@{z} = e^{- z}		cosh(z)- sinh(z) = exp(- z)	Cosh[z]- Sinh[z] == Exp[- z]	Successful	Successful	-	Successful [Tested: 7]
4.28.E4	${\displaystyle{\displaystyle\tanh z=\frac{\sinh z}{\cosh z}}}$ \tanh@@{z} = \frac{\sinh@@{z}}{\cosh@@{z}}		tanh(z) = (sinh(z))/(cosh(z))	Tanh[z] == Divide[Sinh[z],Cosh[z]]	Successful	Successful	-	Successful [Tested: 7]
4.28.E5	${\displaystyle{\displaystyle\operatorname{csch}z=\frac{1}{\sinh z}}}$ \csch@@{z} = \frac{1}{\sinh@@{z}}		csch(z) = (1)/(sinh(z))	Csch[z] == Divide[1,Sinh[z]]	Successful	Successful	-	Successful [Tested: 7]
4.28.E6	${\displaystyle{\displaystyle\operatorname{sech}z=\frac{1}{\cosh z}}}$ \sech@@{z} = \frac{1}{\cosh@@{z}}		sech(z) = (1)/(cosh(z))	Sech[z] == Divide[1,Cosh[z]]	Successful	Successful	-	Successful [Tested: 7]
4.28.E7	${\displaystyle{\displaystyle\coth z=\frac{1}{\tanh z}}}$ \coth@@{z} = \frac{1}{\tanh@@{z}}		coth(z) = (1)/(tanh(z))	Coth[z] == Divide[1,Tanh[z]]	Successful	Successful	-	Successful [Tested: 7]
4.28.E8	${\displaystyle{\displaystyle\sin\left(iz\right)=i\sinh z}}$ \sin@{iz} = i\sinh@@{z}		sin(I*z) = I*sinh(z)	Sin[I*z] == I*Sinh[z]	Successful	Successful	-	Successful [Tested: 7]
4.28.E9	${\displaystyle{\displaystyle\cos\left(iz\right)=\cosh z}}$ \cos@{iz} = \cosh@@{z}		cos(I*z) = cosh(z)	Cos[I*z] == Cosh[z]	Successful	Successful	-	Successful [Tested: 7]
4.28.E10	${\displaystyle{\displaystyle\tan\left(iz\right)=i\tanh z}}$ \tan@{iz} = i\tanh@@{z}		tan(I*z) = I*tanh(z)	Tan[I*z] == I*Tanh[z]	Successful	Successful	-	Successful [Tested: 7]
4.28.E11	${\displaystyle{\displaystyle\csc\left(iz\right)=-i\operatorname{csch}z}}$ \csc@{iz} = -i\csch@@{z}		csc(I*z) = - I*csch(z)	Csc[I*z] == - I*Csch[z]	Successful	Successful	-	Successful [Tested: 7]
4.28.E12	${\displaystyle{\displaystyle\sec\left(iz\right)=\operatorname{sech}z}}$ \sec@{iz} = \sech@@{z}		sec(I*z) = sech(z)	Sec[I*z] == Sech[z]	Successful	Successful	-	Successful [Tested: 7]
4.28.E13	${\displaystyle{\displaystyle\cot\left(iz\right)=-i\coth z}}$ \cot@{iz} = -i\coth@@{z}		cot(I*z) = - I*coth(z)	Cot[I*z] == - I*Coth[z]	Successful	Successful	-	Successful [Tested: 7]
4.31.E1	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{\sinh z}{z}=1}}$ \lim_{z\to 0}\frac{\sinh@@{z}}{z} = 1		limit((sinh(z))/(z), z = 0) = 1	Limit[Divide[Sinh[z],z], z -> 0, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 1]
4.31.E2	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{\tanh z}{z}=1}}$ \lim_{z\to 0}\frac{\tanh@@{z}}{z} = 1		limit((tanh(z))/(z), z = 0) = 1	Limit[Divide[Tanh[z],z], z -> 0, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 1]
4.31.E3	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}}}$ \lim_{z\to 0}\frac{\cosh@@{z}-1}{z^{2}} = \frac{1}{2}		limit((cosh(z)- 1)/((z)^(2)), z = 0) = (1)/(2)	Limit[Divide[Cosh[z]- 1,(z)^(2)], z -> 0, GenerateConditions->None] == Divide[1,2]	Successful	Successful	-	Successful [Tested: 1]
4.32.E1	${\displaystyle{\displaystyle\cosh x\leq\left(\frac{\sinh x}{x}\right)^{3}}}$ \cosh@@{x} \leq \left(\frac{\sinh@@{x}}{x}\right)^{3}		cosh(x) <= ((sinh(x))/(x))^(3)	Cosh[x] <= (Divide[Sinh[x],x])^(3)	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.32.E2	${\displaystyle{\displaystyle\sin x\cos x<\tanh x}}$ \sin@@{x}\cos@@{x} < \tanh@@{x}	${\displaystyle{\displaystyle x>0}}$	sin(x)*cos(x) < tanh(x)	Sin[x]*Cos[x] < Tanh[x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.32.E2	${\displaystyle{\displaystyle\tanh x<x}}$ \tanh@@{x} < x	${\displaystyle{\displaystyle x>0}}$	tanh(x) < x	Tanh[x] < x	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.32.E3	${\displaystyle{\displaystyle|\cosh x-\cosh y|\geq|x-y|\sqrt{\sinh x\sinh y}}}$ |\cosh@@{x}-\cosh@@{y}| \geq |x-y|\sqrt{\sinh@@{x}\sinh@@{y}}	${\displaystyle{\displaystyle x>0,y>0}}$	abs(cosh(x)- cosh(y)) >= abs(x - y)*sqrt(sinh(x)*sinh(y))	Abs[Cosh[x]- Cosh[y]] >= Abs[x - y]*Sqrt[Sinh[x]*Sinh[y]]	Failure	Failure	Successful [Tested: 9]	Successful [Tested: 9]
4.32.E4	${\displaystyle{\displaystyle\operatorname{arctan}x\leq\tfrac{1}{2}\pi\tanh x}}$ \atan@@{x} \leq \tfrac{1}{2}\pi\tanh@@{x}	${\displaystyle{\displaystyle x\geq 0}}$	arctan(x) <= (1)/(2)*Pi*tanh(x)	ArcTan[x] <= Divide[1,2]*Pi*Tanh[x]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
4.34.E1	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sinh z=\cosh z}}$ \deriv{}{z}\sinh@@{z} = \cosh@@{z}		diff(sinh(z), z) = cosh(z)	D[Sinh[z], z] == Cosh[z]	Successful	Successful	-	Successful [Tested: 7]
4.34.E2	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cosh z=\sinh z}}$ \deriv{}{z}\cosh@@{z} = \sinh@@{z}		diff(cosh(z), z) = sinh(z)	D[Cosh[z], z] == Sinh[z]	Successful	Successful	-	Successful [Tested: 7]
4.34.E3	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tanh z={\operatorname{sech}^{2}}z}}$ \deriv{}{z}\tanh@@{z} = \sech^{2}@@{z}		diff(tanh(z), z) = (sech(z))^(2)	D[Tanh[z], z] == (Sech[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.34.E4	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{csch}z=-\operatorname{csch}z\coth z}}$ \deriv{}{z}\csch@@{z} = -\csch@@{z}\coth@@{z}		diff(csch(z), z) = - csch(z)*coth(z)	D[Csch[z], z] == - Csch[z]*Coth[z]	Successful	Successful	-	Successful [Tested: 7]
4.34.E5	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sech}z=-\operatorname{sech}z\tanh z}}$ \deriv{}{z}\sech@@{z} = -\sech@@{z}\tanh@@{z}		diff(sech(z), z) = - sech(z)*tanh(z)	D[Sech[z], z] == - Sech[z]*Tanh[z]	Successful	Successful	-	Successful [Tested: 7]
4.34.E6	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\coth z=-{\operatorname{csch}^{2}}z}}$ \deriv{}{z}\coth@@{z} = -\csch^{2}@@{z}		diff(coth(z), z) = - (csch(z))^(2)	D[Coth[z], z] == - (Csch[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.34.E7	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w=0}}$ \deriv[2]{w}{z}-a^{2}w = 0		diff(w, [z$(2)])- (a)^(2)* w = 0	D[w, {z, 2}]- (a)^(2)* w == 0	Failure	Failure	Failed [300 / 300] Result: -1.948557159-1.125000000*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -1.948557159-1.125000000*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: -1.948557159-1.125000000*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: -1.948557159-1.125000000*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-1.948557158514987, -1.1249999999999998] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.948557158514987, -1.1249999999999998] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.34.E8	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=1}}$ \left(\deriv{w}{z}\right)^{2}-a^{2}w^{2} = 1		(diff(w, z))^(2)- (a)^(2)* (w)^(2) = 1	(D[w, z])^(2)- (a)^(2)* (w)^(2) == 1	Failure	Failure	Failed [300 / 300] Result: -2.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -2.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: -2.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: -2.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-2.125, -1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.125, -1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.34.E9	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=-1}}$ \left(\deriv{w}{z}\right)^{2}-a^{2}w^{2} = -1		(diff(w, z))^(2)- (a)^(2)* (w)^(2) = - 1	(D[w, z])^(2)- (a)^(2)* (w)^(2) == - 1	Failure	Failure	Failed [272 / 300] Result: -.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: -.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: -.125000001-1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [272 / 300] Result: Complex[-0.12500000000000022, -1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.12500000000000022, -1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.34.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}=1}}$ \deriv{w}{z}+a^{2}w^{2} = 1		diff(w, z)+ (a)^(2)* (w)^(2) = 1	D[w, z]+ (a)^(2)* (w)^(2) == 1	Failure	Failure	Failed [272 / 300] Result: .125000001+1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .125000001+1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: .125000001+1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: .125000001+1.948557159*I Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [272 / 300] Result: Complex[0.12500000000000022, 1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.12500000000000022, 1.9485571585149868] Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.34.E11	${\displaystyle{\displaystyle w=A\cosh\left(az\right)+B\sinh\left(az\right)}}$ w = A\cosh@{az}+B\sinh@{az}		w = A*cosh(a*z)+ B*sinh(a*z)	w == A*Cosh[a*z]+ B*Sinh[a*z]	Failure	Failure	Failed [300 / 300] Result: .6001928989+.561234643*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -.6457530113+1.981963256*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: .9837329493+.425340516e-1*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: -.2074648399-3.005064943*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.6001928983405861, 0.5612346426489729] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.645753012062901, 1.9819632558589868] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.34.E12	${\displaystyle{\displaystyle w=(1/a)\sinh\left(az+c\right)}}$ w = (1/a)\sinh@{az+c}		w = (1/a)*sinh(a*z + c)	w == (1/a)*Sinh[a*z + c]	Failure	Failure	Failed [300 / 300] Result: -3.126061208-3.246674013*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .7188715257-.3314459800*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: .265391293e-1+3.580357057*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: .7673365303+.9636329126*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-3.126061206522873, -3.246674011194613] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.7188715253469982, -0.33144598009263954] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.34.E13	${\displaystyle{\displaystyle w=(1/a)\cosh\left(az+c\right)}}$ w = (1/a)\cosh@{az+c}		w = (1/a)*cosh(a*z + c)	w == (1/a)*Cosh[a*z + c]	Failure	Failure	Failed [300 / 300] Result: 4.887803259+4.219013756*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: 1.097709449+1.028092043*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: 1.724372908-2.512669644*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: 1.363701096+.4080617947*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[4.887803257491119, 4.219013753952423] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.0977094487385304, 1.0280920432224616] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.34.E14	${\displaystyle{\displaystyle w=(1/a)\coth\left(az+c\right)}}$ w = (1/a)\coth@{az+c}		w = (1/a)*coth(a*z + c)	w == (1/a)*Coth[a*z + c]	Failure	Failure	Failed [300 / 300] Result: .1990274306+.5049301211*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: .4235738270+.6074604561*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: .2119596261+.4924838498*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: .5938323036-.1576784256*I Test Values: {a = -1.5, c = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.19902743024251868, 0.504930121080845] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.423573826800421, 0.6074604562830159] Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.35.E1	${\displaystyle{\displaystyle\sinh\left(u+v\right)=\sinh u\cosh v+\cosh u\sinh v}}$ \sinh@{u+ v} = \sinh@@{u}\cosh@@{v}+\cosh@@{u}\sinh@@{v}		sinh(u + v) = sinh(u)*cosh(v)+ cosh(u)*sinh(v)	Sinh[u + v] == Sinh[u]*Cosh[v]+ Cosh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E1	${\displaystyle{\displaystyle\sinh\left(u-v\right)=\sinh u\cosh v-\cosh u\sinh v}}$ \sinh@{u- v} = \sinh@@{u}\cosh@@{v}-\cosh@@{u}\sinh@@{v}		sinh(u - v) = sinh(u)*cosh(v)- cosh(u)*sinh(v)	Sinh[u - v] == Sinh[u]*Cosh[v]- Cosh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E2	${\displaystyle{\displaystyle\cosh\left(u+v\right)=\cosh u\cosh v+\sinh u\sinh v}}$ \cosh@{u+ v} = \cosh@@{u}\cosh@@{v}+\sinh@@{u}\sinh@@{v}		cosh(u + v) = cosh(u)*cosh(v)+ sinh(u)*sinh(v)	Cosh[u + v] == Cosh[u]*Cosh[v]+ Sinh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E2	${\displaystyle{\displaystyle\cosh\left(u-v\right)=\cosh u\cosh v-\sinh u\sinh v}}$ \cosh@{u- v} = \cosh@@{u}\cosh@@{v}-\sinh@@{u}\sinh@@{v}		cosh(u - v) = cosh(u)*cosh(v)- sinh(u)*sinh(v)	Cosh[u - v] == Cosh[u]*Cosh[v]- Sinh[u]*Sinh[v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E3	${\displaystyle{\displaystyle\tanh\left(u+v\right)=\frac{\tanh u+\tanh v}{1+\tanh u\tanh v}}}$ \tanh@{u+ v} = \frac{\tanh@@{u}+\tanh@@{v}}{1+\tanh@@{u}\tanh@@{v}}		tanh(u + v) = (tanh(u)+ tanh(v))/(1 + tanh(u)*tanh(v))	Tanh[u + v] == Divide[Tanh[u]+ Tanh[v],1 + Tanh[u]*Tanh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E3	${\displaystyle{\displaystyle\tanh\left(u-v\right)=\frac{\tanh u-\tanh v}{1-\tanh u\tanh v}}}$ \tanh@{u- v} = \frac{\tanh@@{u}-\tanh@@{v}}{1-\tanh@@{u}\tanh@@{v}}		tanh(u - v) = (tanh(u)- tanh(v))/(1 - tanh(u)*tanh(v))	Tanh[u - v] == Divide[Tanh[u]- Tanh[v],1 - Tanh[u]*Tanh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E4	${\displaystyle{\displaystyle\coth\left(u+v\right)=\frac{+\coth u\coth v+1}{\coth u+\coth v}}}$ \coth@{u+ v} = \frac{+\coth@@{u}\coth@@{v}+1}{\coth@@{u}+\coth@@{v}}		coth(u + v) = (+ coth(u)*coth(v)+ 1)/(coth(u)+ coth(v))	Coth[u + v] == Divide[+ Coth[u]*Coth[v]+ 1,Coth[u]+ Coth[v]]	Successful	Successful	-	Failed [10 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: Complex[4.333014420201075*^14, -2.3525621062227262*^14] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.35.E4	${\displaystyle{\displaystyle\coth\left(u-v\right)=\frac{-\coth u\coth v+1}{\coth u-\coth v}}}$ \coth@{u- v} = \frac{-\coth@@{u}\coth@@{v}+1}{\coth@@{u}-\coth@@{v}}		coth(u - v) = (- coth(u)*coth(v)+ 1)/(coth(u)- coth(v))	Coth[u - v] == Divide[- Coth[u]*Coth[v]+ 1,Coth[u]- Coth[v]]	Successful	Successful	-	Failed [10 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.35.E5	${\displaystyle{\displaystyle\sinh u+\sinh v=2\sinh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right)}}$ \sinh@@{u}+\sinh@@{v} = 2\sinh@{\frac{u+v}{2}}\cosh@{\frac{u-v}{2}}		sinh(u)+ sinh(v) = 2*sinh((u + v)/(2))*cosh((u - v)/(2))	Sinh[u]+ Sinh[v] == 2*Sinh[Divide[u + v,2]]*Cosh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E6	${\displaystyle{\displaystyle\sinh u-\sinh v=2\cosh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right)}}$ \sinh@@{u}-\sinh@@{v} = 2\cosh@{\frac{u+v}{2}}\sinh@{\frac{u-v}{2}}		sinh(u)- sinh(v) = 2*cosh((u + v)/(2))*sinh((u - v)/(2))	Sinh[u]- Sinh[v] == 2*Cosh[Divide[u + v,2]]*Sinh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E7	${\displaystyle{\displaystyle\cosh u+\cosh v=2\cosh\left(\frac{u+v}{2}\right)\cosh\left(\frac{u-v}{2}\right)}}$ \cosh@@{u}+\cosh@@{v} = 2\cosh@{\frac{u+v}{2}}\cosh@{\frac{u-v}{2}}		cosh(u)+ cosh(v) = 2*cosh((u + v)/(2))*cosh((u - v)/(2))	Cosh[u]+ Cosh[v] == 2*Cosh[Divide[u + v,2]]*Cosh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E8	${\displaystyle{\displaystyle\cosh u-\cosh v=2\sinh\left(\frac{u+v}{2}\right)\sinh\left(\frac{u-v}{2}\right)}}$ \cosh@@{u}-\cosh@@{v} = 2\sinh@{\frac{u+v}{2}}\sinh@{\frac{u-v}{2}}		cosh(u)- cosh(v) = 2*sinh((u + v)/(2))*sinh((u - v)/(2))	Cosh[u]- Cosh[v] == 2*Sinh[Divide[u + v,2]]*Sinh[Divide[u - v,2]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E9	${\displaystyle{\displaystyle\tanh u+\tanh v=\frac{\sinh\left(u+v\right)}{\cosh u\cosh v}}}$ \tanh@@{u}+\tanh@@{v} = \frac{\sinh@{u+ v}}{\cosh@@{u}\cosh@@{v}}		tanh(u)+ tanh(v) = (sinh(u + v))/(cosh(u)*cosh(v))	Tanh[u]+ Tanh[v] == Divide[Sinh[u + v],Cosh[u]*Cosh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E9	${\displaystyle{\displaystyle\tanh u-\tanh v=\frac{\sinh\left(u-v\right)}{\cosh u\cosh v}}}$ \tanh@@{u}-\tanh@@{v} = \frac{\sinh@{u- v}}{\cosh@@{u}\cosh@@{v}}		tanh(u)- tanh(v) = (sinh(u - v))/(cosh(u)*cosh(v))	Tanh[u]- Tanh[v] == Divide[Sinh[u - v],Cosh[u]*Cosh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E10	${\displaystyle{\displaystyle\coth u+\coth v=\frac{\sinh\left(v+u\right)}{\sinh u\sinh v}}}$ \coth@@{u}+\coth@@{v} = \frac{\sinh@{v+ u}}{\sinh@@{u}\sinh@@{v}}		coth(u)+ coth(v) = (sinh(v + u))/(sinh(u)*sinh(v))	Coth[u]+ Coth[v] == Divide[Sinh[v + u],Sinh[u]*Sinh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E10	${\displaystyle{\displaystyle\coth u-\coth v=\frac{\sinh\left(v-u\right)}{\sinh u\sinh v}}}$ \coth@@{u}-\coth@@{v} = \frac{\sinh@{v- u}}{\sinh@@{u}\sinh@@{v}}		coth(u)- coth(v) = (sinh(v - u))/(sinh(u)*sinh(v))	Coth[u]- Coth[v] == Divide[Sinh[v - u],Sinh[u]*Sinh[v]]	Successful	Successful	-	Successful [Tested: 100]
4.35.E11	${\displaystyle{\displaystyle{\cosh^{2}}z-{\sinh^{2}}z=1}}$ \cosh^{2}@@{z}-\sinh^{2}@@{z} = 1		(cosh(z))^(2)- (sinh(z))^(2) = 1	(Cosh[z])^(2)- (Sinh[z])^(2) == 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E12	${\displaystyle{\displaystyle{\operatorname{sech}^{2}}z=1-{\tanh^{2}}z}}$ \sech^{2}@@{z} = 1-\tanh^{2}@@{z}		(sech(z))^(2) = 1 - (tanh(z))^(2)	(Sech[z])^(2) == 1 - (Tanh[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.35.E13	${\displaystyle{\displaystyle{\operatorname{csch}^{2}}z={\coth^{2}}z-1}}$ \csch^{2}@@{z} = \coth^{2}@@{z}-1		(csch(z))^(2) = (coth(z))^(2)- 1	(Csch[z])^(2) == (Coth[z])^(2)- 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E14	${\displaystyle{\displaystyle 2\sinh u\sinh v=\cosh\left(u+v\right)-\cosh\left(u-v\right)}}$ 2\sinh@@{u}\sinh@@{v} = \cosh@{u+v}-\cosh@{u-v}		2*sinh(u)*sinh(v) = cosh(u + v)- cosh(u - v)	2*Sinh[u]*Sinh[v] == Cosh[u + v]- Cosh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E15	${\displaystyle{\displaystyle 2\cosh u\cosh v=\cosh\left(u+v\right)+\cosh\left(u-v\right)}}$ 2\cosh@@{u}\cosh@@{v} = \cosh@{u+v}+\cosh@{u-v}		2*cosh(u)*cosh(v) = cosh(u + v)+ cosh(u - v)	2*Cosh[u]*Cosh[v] == Cosh[u + v]+ Cosh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E16	${\displaystyle{\displaystyle 2\sinh u\cosh v=\sinh\left(u+v\right)+\sinh\left(u-v\right)}}$ 2\sinh@@{u}\cosh@@{v} = \sinh@{u+v}+\sinh@{u-v}		2*sinh(u)*cosh(v) = sinh(u + v)+ sinh(u - v)	2*Sinh[u]*Cosh[v] == Sinh[u + v]+ Sinh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E17	${\displaystyle{\displaystyle{\sinh^{2}}u-{\sinh^{2}}v=\sinh\left(u+v\right)\sinh\left(u-v\right)}}$ \sinh^{2}@@{u}-\sinh^{2}@@{v} = \sinh@{u+v}\sinh@{u-v}		(sinh(u))^(2)- (sinh(v))^(2) = sinh(u + v)*sinh(u - v)	(Sinh[u])^(2)- (Sinh[v])^(2) == Sinh[u + v]*Sinh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E18	${\displaystyle{\displaystyle{\cosh^{2}}u-{\cosh^{2}}v=\sinh\left(u+v\right)\sinh\left(u-v\right)}}$ \cosh^{2}@@{u}-\cosh^{2}@@{v} = \sinh@{u+v}\sinh@{u-v}		(cosh(u))^(2)- (cosh(v))^(2) = sinh(u + v)*sinh(u - v)	(Cosh[u])^(2)- (Cosh[v])^(2) == Sinh[u + v]*Sinh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E19	${\displaystyle{\displaystyle{\sinh^{2}}u+{\cosh^{2}}v=\cosh\left(u+v\right)\cosh\left(u-v\right)}}$ \sinh^{2}@@{u}+\cosh^{2}@@{v} = \cosh@{u+v}\cosh@{u-v}		(sinh(u))^(2)+ (cosh(v))^(2) = cosh(u + v)*cosh(u - v)	(Sinh[u])^(2)+ (Cosh[v])^(2) == Cosh[u + v]*Cosh[u - v]	Successful	Successful	-	Successful [Tested: 100]
4.35.E20	${\displaystyle{\displaystyle\sinh\frac{z}{2}=\left(\frac{\cosh z-1}{2}\right)^{1/2}}}$ \sinh@@{\frac{z}{2}} = \left(\frac{\cosh@@{z}-1}{2}\right)^{1/2}		sinh((z)/(2)) = ((cosh(z)- 1)/(2))^(1/2)	Sinh[Divide[z,2]] == (Divide[Cosh[z]- 1,2])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.4585952894+.8655770340*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: -.8655716642-.5419255224*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[-0.4585952893468803, 0.8655770337160631] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8655716640572735, -0.5419255224573363] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.35.E21	${\displaystyle{\displaystyle\cosh\frac{z}{2}=\left(\frac{\cosh z+1}{2}\right)^{1/2}}}$ \cosh@@{\frac{z}{2}} = \left(\frac{\cosh@@{z}+1}{2}\right)^{1/2}		cosh((z)/(2)) = ((cosh(z)+ 1)/(2))^(1/2)	Cosh[Divide[z,2]] == (Divide[Cosh[z]+ 1,2])^(1/2)	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.35.E22	${\displaystyle{\displaystyle\tanh\frac{z}{2}=\left(\frac{\cosh z-1}{\cosh z+1}\right)^{1/2}}}$ \tanh@@{\frac{z}{2}} = \left(\frac{\cosh@@{z}-1}{\cosh@@{z}+1}\right)^{1/2}		tanh((z)/(2)) = ((cosh(z)- 1)/(cosh(z)+ 1))^(1/2)	Tanh[Divide[z,2]] == (Divide[Cosh[z]- 1,Cosh[z]+ 1])^(1/2)	Failure	Failure	Failed [2 / 7] Result: -.5869891489+.8580864930*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: -.8595320616-.4211742148*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[-0.5869891488727425, 0.858086492859854] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.8595320613685857, -0.42117421488499707] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.35.E22	${\displaystyle{\displaystyle\left(\frac{\cosh z-1}{\cosh z+1}\right)^{1/2}=\frac{\cosh z-1}{\sinh z}}}$ \left(\frac{\cosh@@{z}-1}{\cosh@@{z}+1}\right)^{1/2} = \frac{\cosh@@{z}-1}{\sinh@@{z}}		((cosh(z)- 1)/(cosh(z)+ 1))^(1/2) = (cosh(z)- 1)/(sinh(z))	(Divide[Cosh[z]- 1,Cosh[z]+ 1])^(1/2) == Divide[Cosh[z]- 1,Sinh[z]]	Failure	Failure	Failed [2 / 7] Result: .5869891489-.8580864930*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: .8595320615+.4211742148*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[0.5869891488727426, -0.8580864928598539] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.859532061368586, 0.42117421488499684] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
4.35.E22	${\displaystyle{\displaystyle\frac{\cosh z-1}{\sinh z}=\frac{\sinh z}{\cosh z+1}}}$ \frac{\cosh@@{z}-1}{\sinh@@{z}} = \frac{\sinh@@{z}}{\cosh@@{z}+1}		(cosh(z)- 1)/(sinh(z)) = (sinh(z))/(cosh(z)+ 1)	Divide[Cosh[z]- 1,Sinh[z]] == Divide[Sinh[z],Cosh[z]+ 1]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
4.35.E23	${\displaystyle{\displaystyle\sinh\left(-z\right)=-\sinh z}}$ \sinh@{-z} = -\sinh@@{z}		sinh(- z) = - sinh(z)	Sinh[- z] == - Sinh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E24	${\displaystyle{\displaystyle\cosh\left(-z\right)=\cosh z}}$ \cosh@{-z} = \cosh@@{z}		cosh(- z) = cosh(z)	Cosh[- z] == Cosh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E25	${\displaystyle{\displaystyle\tanh\left(-z\right)=-\tanh z}}$ \tanh@{-z} = -\tanh@@{z}		tanh(- z) = - tanh(z)	Tanh[- z] == - Tanh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E26	${\displaystyle{\displaystyle\sinh\left(2z\right)=2\sinh z\cosh z}}$ \sinh@{2z} = 2\sinh@@{z}\cosh@@{z}		sinh(2*z) = 2*sinh(z)*cosh(z)	Sinh[2*z] == 2*Sinh[z]*Cosh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E26	${\displaystyle{\displaystyle 2\sinh z\cosh z=\frac{2\tanh z}{1-{\tanh^{2}}z}}}$ 2\sinh@@{z}\cosh@@{z} = \frac{2\tanh@@{z}}{1-\tanh^{2}@@{z}}		2*sinh(z)*cosh(z) = (2*tanh(z))/(1 - (tanh(z))^(2))	2*Sinh[z]*Cosh[z] == Divide[2*Tanh[z],1 - (Tanh[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.35.E27	${\displaystyle{\displaystyle\cosh\left(2z\right)=2{\cosh^{2}}z-1}}$ \cosh@{2z} = 2\cosh^{2}@@{z}-1		cosh(2*z) = 2*(cosh(z))^(2)- 1	Cosh[2*z] == 2*(Cosh[z])^(2)- 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E27	${\displaystyle{\displaystyle 2{\cosh^{2}}z-1=2{\sinh^{2}}z+1\\ }}$ 2\cosh^{2}@@{z}-1 = 2\sinh^{2}@@{z}+1\\		2*(cosh(z))^(2)- 1 = 2*(sinh(z))^(2)+ 1	2*(Cosh[z])^(2)- 1 == 2*(Sinh[z])^(2)+ 1	Successful	Successful	-	Successful [Tested: 7]
4.35.E27	${\displaystyle{\displaystyle 2{\sinh^{2}}z+1\\ ={\cosh^{2}}z+{\sinh^{2}}z}}$ 2\sinh^{2}@@{z}+1\\ = \cosh^{2}@@{z}+\sinh^{2}@@{z}		2*(sinh(z))^(2)+ 1 = (cosh(z))^(2)+ (sinh(z))^(2)	2*(Sinh[z])^(2)+ 1 == (Cosh[z])^(2)+ (Sinh[z])^(2)	Successful	Successful	-	Successful [Tested: 7]
4.35.E28	${\displaystyle{\displaystyle\tanh\left(2z\right)=\frac{2\tanh z}{1+{\tanh^{2}}z}}}$ \tanh@{2z} = \frac{2\tanh@@{z}}{1+\tanh^{2}@@{z}}		tanh(2*z) = (2*tanh(z))/(1 + (tanh(z))^(2))	Tanh[2*z] == Divide[2*Tanh[z],1 + (Tanh[z])^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.35.E29	${\displaystyle{\displaystyle\sinh\left(3z\right)=3\sinh z+4{\sinh^{3}}z}}$ \sinh@{3z} = 3\sinh@@{z}+4\sinh^{3}@@{z}		sinh(3*z) = 3*sinh(z)+ 4*(sinh(z))^(3)	Sinh[3*z] == 3*Sinh[z]+ 4*(Sinh[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.35.E30	${\displaystyle{\displaystyle\cosh\left(3z\right)=-3\cosh z+4{\cosh^{3}}z}}$ \cosh@{3z} = -3\cosh@@{z}+4\cosh^{3}@@{z}		cosh(3*z) = - 3*cosh(z)+ 4*(cosh(z))^(3)	Cosh[3*z] == - 3*Cosh[z]+ 4*(Cosh[z])^(3)	Successful	Successful	-	Successful [Tested: 7]
4.35.E31	${\displaystyle{\displaystyle\sinh\left(4z\right)=4{\sinh^{3}}z\cosh z+4{\cosh^{3}}z\sinh z}}$ \sinh@{4z} = 4\sinh^{3}@@{z}\cosh@@{z}+4\cosh^{3}@@{z}\sinh@@{z}		sinh(4*z) = 4*(sinh(z))^(3)* cosh(z)+ 4*(cosh(z))^(3)* sinh(z)	Sinh[4*z] == 4*(Sinh[z])^(3)* Cosh[z]+ 4*(Cosh[z])^(3)* Sinh[z]	Successful	Successful	-	Successful [Tested: 7]
4.35.E32	${\displaystyle{\displaystyle\cosh\left(4z\right)={\cosh^{4}}z+6{\sinh^{2}}z{\cosh^{2}}z+{\sinh^{4}}z}}$ \cosh@{4z} = \cosh^{4}@@{z}+6\sinh^{2}@@{z}\cosh^{2}@@{z}+\sinh^{4}@@{z}		cosh(4*z) = (cosh(z))^(4)+ 6*(sinh(z))^(2)* (cosh(z))^(2)+ (sinh(z))^(4)	Cosh[4*z] == (Cosh[z])^(4)+ 6*(Sinh[z])^(2)* (Cosh[z])^(2)+ (Sinh[z])^(4)	Successful	Successful	-	Successful [Tested: 7]
4.35.E33	${\displaystyle{\displaystyle\cosh\left(nz\right)+\sinh\left(nz\right)=(\cosh z+\sinh z)^{n}}}$ \cosh@{nz}+\sinh@{nz} = (\cosh@@{z}+\sinh@@{z})^{n}		cosh(n*z)+ sinh(n*z) = (cosh(z)+ sinh(z))^(n)	Cosh[n*z]+ Sinh[n*z] == (Cosh[z]+ Sinh[z])^(n)	Successful	Successful	-	Successful [Tested: 7]
4.35.E33	${\displaystyle{\displaystyle\cosh\left(nz\right)-\sinh\left(nz\right)=(\cosh z-\sinh z)^{n}}}$ \cosh@{nz}-\sinh@{nz} = (\cosh@@{z}-\sinh@@{z})^{n}		cosh(n*z)- sinh(n*z) = (cosh(z)- sinh(z))^(n)	Cosh[n*z]- Sinh[n*z] == (Cosh[z]- Sinh[z])^(n)	Successful	Successful	-	Successful [Tested: 7]
4.35.E34	${\displaystyle{\displaystyle\sinh z=\sinh x\cos y+i\cosh x\sin y}}$ \sinh@@{z} = \sinh@@{x}\cos@@{y}+i\cosh@@{x}\sin@@{y}		sinh(x + y*I) = sinh(x)*cos(y)+ I*cosh(x)*sin(y)	Sinh[x + y*I] == Sinh[x]*Cos[y]+ I*Cosh[x]*Sin[y]	Successful	Successful	-	Successful [Tested: 18]
4.35.E35	${\displaystyle{\displaystyle\cosh z=\cosh x\cos y+i\sinh x\sin y}}$ \cosh@@{z} = \cosh@@{x}\cos@@{y}+i\sinh@@{x}\sin@@{y}		cosh(x + y*I) = cosh(x)*cos(y)+ I*sinh(x)*sin(y)	Cosh[x + y*I] == Cosh[x]*Cos[y]+ I*Sinh[x]*Sin[y]	Successful	Successful	-	Successful [Tested: 18]
4.35.E36	${\displaystyle{\displaystyle\tanh z=\frac{\sinh\left(2x\right)+i\sin\left(2y\right)}{\cosh\left(2x\right)+\cos\left(2y\right)}}}$ \tanh@@{z} = \frac{\sinh@{2x}+i\sin@{2y}}{\cosh@{2x}+\cos@{2y}}		tanh(x + y*I) = (sinh(2*x)+ I*sin(2*y))/(cosh(2*x)+ cos(2*y))	Tanh[x + y*I] == Divide[Sinh[2*x]+ I*Sin[2*y],Cosh[2*x]+ Cos[2*y]]	Successful	Successful	-	Successful [Tested: 18]
4.35.E37	${\displaystyle{\displaystyle\coth z=\frac{\sinh\left(2x\right)-i\sin\left(2y\right)}{\cosh\left(2x\right)-\cos\left(2y\right)}}}$ \coth@@{z} = \frac{\sinh@{2x}-i\sin@{2y}}{\cosh@{2x}-\cos@{2y}}		coth(x + y*I) = (sinh(2*x)- I*sin(2*y))/(cosh(2*x)- cos(2*y))	Coth[x + y*I] == Divide[Sinh[2*x]- I*Sin[2*y],Cosh[2*x]- Cos[2*y]]	Successful	Successful	-	Successful [Tested: 18]
4.35.E38	${\displaystyle{\displaystyle|\sinh z|=({\sinh^{2}}x+{\sin^{2}}y)^{1/2}}}$ |\sinh@@{z}| = (\sinh^{2}@@{x}+\sin^{2}@@{y})^{1/2}		abs(sinh(x + y*I)) = ((sinh(x))^(2)+ (sin(y))^(2))^(1/2)	Abs[Sinh[x + y*I]] == ((Sinh[x])^(2)+ (Sin[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.35.E38	${\displaystyle{\displaystyle({\sinh^{2}}x+{\sin^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x\right)-\cos\left(2y\right))\right)^{1/2}}}$ (\sinh^{2}@@{x}+\sin^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}(\cosh@{2x}-\cos@{2y})\right)^{1/2}		((sinh(x))^(2)+ (sin(y))^(2))^(1/2) = ((1)/(2)*(cosh(2*x)- cos(2*y)))^(1/2)	((Sinh[x])^(2)+ (Sin[y])^(2))^(1/2) == (Divide[1,2]*(Cosh[2*x]- Cos[2*y]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.35.E39	${\displaystyle{\displaystyle|\cosh z|=({\sinh^{2}}x+{\cos^{2}}y)^{1/2}}}$ |\cosh@@{z}| = (\sinh^{2}@@{x}+\cos^{2}@@{y})^{1/2}		abs(cosh(x + y*I)) = ((sinh(x))^(2)+ (cos(y))^(2))^(1/2)	Abs[Cosh[x + y*I]] == ((Sinh[x])^(2)+ (Cos[y])^(2))^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.35.E39	${\displaystyle{\displaystyle({\sinh^{2}}x+{\cos^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2x\right)+\cos\left(2y\right))\right)^{1/2}}}$ (\sinh^{2}@@{x}+\cos^{2}@@{y})^{1/2} = \left(\tfrac{1}{2}(\cosh@{2x}+\cos@{2y})\right)^{1/2}		((sinh(x))^(2)+ (cos(y))^(2))^(1/2) = ((1)/(2)*(cosh(2*x)+ cos(2*y)))^(1/2)	((Sinh[x])^(2)+ (Cos[y])^(2))^(1/2) == (Divide[1,2]*(Cosh[2*x]+ Cos[2*y]))^(1/2)	Successful	Successful	-	Successful [Tested: 18]
4.35.E40	${\displaystyle{\displaystyle|\tanh z|=\left(\frac{\cosh\left(2x\right)-\cos\left(2y\right)}{\cosh\left(2x\right)+\cos\left(2y\right)}\right)^{1/2}}}$ |\tanh@@{z}| = \left(\frac{\cosh@{2x}-\cos@{2y}}{\cosh@{2x}+\cos@{2y}}\right)^{1/2}		abs(tanh(x + y*I)) = ((cosh(2*x)- cos(2*y))/(cosh(2*x)+ cos(2*y)))^(1/2)	Abs[Tanh[x + y*I]] == (Divide[Cosh[2*x]- Cos[2*y],Cosh[2*x]+ Cos[2*y]])^(1/2)	Successful	Failure	-	Successful [Tested: 18]
4.36.E1	${\displaystyle{\displaystyle\sinh z=z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right)}}$ \sinh@@{z} = z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right)		sinh(z) = z*product(1 +((z)^(2))/((n)^(2)* (Pi)^(2)), n = 1..infinity)	Sinh[z] == z*Product[1 +Divide[(z)^(2),(n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
4.36.E2	${\displaystyle{\displaystyle\cosh z=\prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)}}$ \cosh@@{z} = \prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right)		cosh(z) = product(1 +(4*(z)^(2))/((2*n - 1)^(2)* (Pi)^(2)), n = 1..infinity)	Cosh[z] == Product[1 +Divide[4*(z)^(2),(2*n - 1)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
4.36.E3	${\displaystyle{\displaystyle\coth z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}}}}$ \coth@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}}		coth(z) = (1)/(z)+ 2*z*sum((1)/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)	Coth[z] == Divide[1,z]+ 2*z*Sum[Divide[1,(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.36.E4	${\displaystyle{\displaystyle{\operatorname{csch}^{2}}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}}}}$ \csch^{2}@@{z} = \sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}}		(csch(z))^(2) = sum((1)/((z - n*Pi*I)^(2)), n = - infinity..infinity)	(Csch[z])^(2) == Sum[Divide[1,(z - n*Pi*I)^(2)], {n, - Infinity, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.36.E5	${\displaystyle{\displaystyle\operatorname{csch}z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}}}$ \csch@@{z} = \frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^{2}}		csch(z) = (1)/(z)+ 2*z*sum(((- 1)^(n))/((z)^(2)+ (n)^(2)* (Pi)^(2)), n = 1..infinity)	Csch[z] == Divide[1,z]+ 2*z*Sum[Divide[(- 1)^(n),(z)^(2)+ (n)^(2)* (Pi)^(2)], {n, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.37.E1	${\displaystyle{\displaystyle\operatorname{Arcsinh}z=\int_{0}^{z}\frac{\mathrm{d}t}{(1+t^{2})^{1/2}}}}$ \Asinh@@{z} = \int_{0}^{z}\frac{\diff{t}}{(1+t^{2})^{1/2}}		Error	ArcSinh[z] == Integrate[Divide[1,(1 + (t)^(2))^(1/2)], {t, 0, z}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.37.E2	${\displaystyle{\displaystyle\operatorname{Arccosh}z=\int_{1}^{z}\frac{\mathrm{d}t}{(t^{2}-1)^{1/2}}}}$ \Acosh@@{z} = \int_{1}^{z}\frac{\diff{t}}{(t^{2}-1)^{1/2}}		Error	ArcCosh[z] == Integrate[Divide[1,((t)^(2)- 1)^(1/2)], {t, 1, z}, GenerateConditions->None]	Missing Macro Error	Aborted	-	Skipped - Because timed out
4.37.E3	${\displaystyle{\displaystyle\operatorname{Arctanh}z=\int_{0}^{z}\frac{\mathrm{d}t}{1-t^{2}}}}$ \Atanh@@{z} = \int_{0}^{z}\frac{\diff{t}}{1-t^{2}}		Error	ArcTanh[z] == Integrate[Divide[1,1 - (t)^(2)], {t, 0, z}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 1]
4.37.E4	${\displaystyle{\displaystyle\operatorname{Arccsch}z=\operatorname{Arcsinh}\left(1/z\right)}}$ \Acsch@@{z} = \Asinh@{1/z}		Error	ArcCsch[z] == ArcSinh[1/z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.37.E5	${\displaystyle{\displaystyle\operatorname{Arcsech}z=\operatorname{Arccosh}\left(1/z\right)}}$ \Asech@@{z} = \Acosh@{1/z}		Error	ArcSech[z] == ArcCosh[1/z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.37.E6	${\displaystyle{\displaystyle\operatorname{Arccoth}z=\operatorname{Arctanh}\left(1/z\right)}}$ \Acoth@@{z} = \Atanh@{1/z}		Error	ArcCoth[z] == ArcTanh[1/z]	Missing Macro Error	Successful	-	Successful [Tested: 7]
4.37.E7	${\displaystyle{\displaystyle\operatorname{arccsch}z=\operatorname{arcsinh}\left(1/z\right)}}$ \acsch@@{z} = \asinh@{1/z}		arccsch(z) = arcsinh(1/z)	ArcCsch[z] == ArcSinh[1/z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.37.E8	${\displaystyle{\displaystyle\operatorname{arcsech}z=\operatorname{arccosh}\left(1/z\right)}}$ \asech@@{z} = \acosh@{1/z}		arcsech(z) = arccosh(1/z)	ArcSech[z] == ArcCosh[1/z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.37.E9	${\displaystyle{\displaystyle\operatorname{arccoth}z=\operatorname{arctanh}\left(1/z\right)}}$ \acoth@@{z} = \atanh@{1/z}		arccoth(z) = arctanh(1/z)	ArcCoth[z] == ArcTanh[1/z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 1]
4.37.E10	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(-z\right)=-\operatorname{arcsinh}z}}$ \asinh@{-z} = -\asinh@@{z}		arcsinh(- z) = - arcsinh(z)	ArcSinh[- z] == - ArcSinh[z]	Successful	Successful	-	Successful [Tested: 7]
4.37.E11	${\displaystyle{\displaystyle\operatorname{arccosh}\left(-z\right)=+\pi i+\operatorname{arccosh}z}}$ \acosh@{-z} = +\pi i+\acosh@@{z}		arccosh(- z) = + Pi*I + arccosh(z)	ArcCosh[- z] == + Pi*I + ArcCosh[z]	Failure	Failure	Failed [3 / 7] Result: 0.-6.283185307*I Test Values: {z = 1/2*3^(1/2)+1/2*I, Im(z) = 1/2} Result: 0.-6.283185307*I Test Values: {z = -1/2+1/2*I*3^(1/2), Im(z) = 1/2} Result: -2.094395103*I Test Values: {z = .5, Im(z) = 1/2}	Failed [1 / 1] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[z, Complex[0, Rational[1, 2]]]}
4.37.E11	${\displaystyle{\displaystyle\operatorname{arccosh}\left(-z\right)=-\pi i+\operatorname{arccosh}z}}$ \acosh@{-z} = -\pi i+\acosh@@{z}		arccosh(- z) = - Pi*I + arccosh(z)	ArcCosh[- z] == - Pi*I + ArcCosh[z]	Failure	Failure	Failed [5 / 7] Result: 0.+6.283185307*I Test Values: {z = 1/2-1/2*I*3^(1/2), Im(z) = 1/2} Result: 0.+6.283185307*I Test Values: {z = -1/2*3^(1/2)-1/2*I, Im(z) = 1/2} Result: 0.+6.283185308*I Test Values: {z = 1.5, Im(z) = 1/2} Result: 4.188790205*I Test Values: {z = .5, Im(z) = 1/2} ... skip entries to safe data	Successful [Tested: 1]
4.37.E12	${\displaystyle{\displaystyle\operatorname{arctanh}\left(-z\right)=-\operatorname{arctanh}z}}$ \atanh@{-z} = -\atanh@@{z}		arctanh(- z) = - arctanh(z)	ArcTanh[- z] == - ArcTanh[z]	Successful	Successful	-	Successful [Tested: 1]
4.37.E13	${\displaystyle{\displaystyle\operatorname{arccsch}\left(-z\right)=-\operatorname{arccsch}z}}$ \acsch@{-z} = -\acsch@@{z}		arccsch(- z) = - arccsch(z)	ArcCsch[- z] == - ArcCsch[z]	Successful	Successful	-	Successful [Tested: 7]
4.37.E14	${\displaystyle{\displaystyle\operatorname{arcsech}\left(-z\right)=-\pi i+\operatorname{arcsech}z}}$ \asech@{-z} = -\pi i+\asech@@{z}		arcsech(- z) = - Pi*I + arcsech(z)	ArcSech[- z] == - Pi*I + ArcSech[z]	Failure	Failure	Failed [5 / 7] Result: 0.+6.283185307*I Test Values: {z = 1/2*3^(1/2)+1/2*I, Im(z) = 1/2} Result: 0.+6.283185307*I Test Values: {z = -1/2+1/2*I*3^(1/2), Im(z) = 1/2} Result: 4.601047966*I Test Values: {z = 1.5, Im(z) = 1/2} Result: 0.+6.283185308*I Test Values: {z = .5, Im(z) = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[z, Complex[0, Rational[1, 2]]]}
4.37.E14	${\displaystyle{\displaystyle\operatorname{arcsech}\left(-z\right)=+\pi i+\operatorname{arcsech}z}}$ \asech@{-z} = +\pi i+\asech@@{z}		arcsech(- z) = + Pi*I + arcsech(z)	ArcSech[- z] == + Pi*I + ArcSech[z]	Failure	Failure	Failed [4 / 7] Result: 0.-6.283185307*I Test Values: {z = 1/2-1/2*I*3^(1/2), Im(z) = 1/2} Result: 0.-6.283185307*I Test Values: {z = -1/2*3^(1/2)-1/2*I, Im(z) = 1/2} Result: -1.682137342*I Test Values: {z = 1.5, Im(z) = 1/2} Result: -2.094395103*I Test Values: {z = 2, Im(z) = 1/2}	Successful [Tested: 1]
4.37.E15	${\displaystyle{\displaystyle\operatorname{arccoth}\left(-z\right)=-\operatorname{arccoth}z}}$ \acoth@{-z} = -\acoth@@{z}		arccoth(- z) = - arccoth(z)	ArcCoth[- z] == - ArcCoth[z]	Failure	Successful	Failed [1 / 7] Result: 0.-3.141592654*I Test Values: {z = .5, z = 1/2}	Successful [Tested: 1]
4.37.E16	${\displaystyle{\displaystyle\operatorname{arcsinh}z=\ln\left((z^{2}+1)^{1/2}+z\right)}}$ \asinh@@{z} = \ln@{(z^{2}+1)^{1/2}+z}		arcsinh(z) = ln(((z)^(2)+ 1)^(1/2)+ z)	ArcSinh[z] == Log[((z)^(2)+ 1)^(1/2)+ z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 1]
4.37.E17	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}\pi i+\ln\left((y^{2}-1)^{1/2}+y\right)}}$ \asinh@{iy} = \tfrac{1}{2}\pi i+\ln@{(y^{2}-1)^{1/2}+y}		arcsinh(I*y) = (1)/(2)*Pi*I + ln(((y)^(2)- 1)^(1/2)+ y)	ArcSinh[I*y] == Divide[1,2]*Pi*I + Log[((y)^(2)- 1)^(1/2)+ y]	Failure	Successful	Failed [4 / 6] Result: .7e-9-6.283185308*I Test Values: {y = -1.5, y = 3/2} Result: -.1347500000e-10-4.188790205*I Test Values: {y = -.5, y = 3/2} Result: -.1347500000e-10-2.094395102*I Test Values: {y = .5, y = 3/2} Result: .2e-8-6.283185308*I Test Values: {y = -2, y = 3/2}	Successful [Tested: 1]
4.37.E17	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}\pi i-\ln\left((y^{2}-1)^{1/2}+y\right)}}$ \asinh@{iy} = \tfrac{1}{2}\pi i-\ln@{(y^{2}-1)^{1/2}+y}		arcsinh(I*y) = (1)/(2)*Pi*I - ln(((y)^(2)- 1)^(1/2)+ y)	ArcSinh[I*y] == Divide[1,2]*Pi*I - Log[((y)^(2)- 1)^(1/2)+ y]	Failure	Failure	Failed [4 / 6] Result: -1.924847301+0.*I Test Values: {y = -1.5, y = 3/2} Result: 1.924847300+0.*I Test Values: {y = 1.5, y = 3/2} Result: -2.633915796+0.*I Test Values: {y = -2, y = 3/2} Result: 2.633915794+0.*I Test Values: {y = 2, y = 3/2}	Failed [1 / 1] Result: Complex[1.9248473002384139, 0.0] Test Values: {Rule[y, Rational[3, 2]]}
4.37.E18	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2}\pi i+\ln\left((y^{2}-1)^{1/2}-y\right)}}$ \asinh@{iy} = -\tfrac{1}{2}\pi i+\ln@{(y^{2}-1)^{1/2}-y}		arcsinh(I*y) = -(1)/(2)*Pi*I + ln(((y)^(2)- 1)^(1/2)- y)	ArcSinh[I*y] == -Divide[1,2]*Pi*I + Log[((y)^(2)- 1)^(1/2)- y]	Failure	Failure	Failed [4 / 6] Result: -1.924847300+0.*I Test Values: {y = -1.5, y = -2} Result: 1.924847301+0.*I Test Values: {y = 1.5, y = -2} Result: -2.633915794+0.*I Test Values: {y = -2, y = -2} Result: 2.633915796+0.*I Test Values: {y = 2, y = -2}	Failed [1 / 1] Result: Complex[-2.633915793849633, 0.0] Test Values: {Rule[y, -2]}
4.37.E18	${\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2}\pi i-\ln\left((y^{2}-1)^{1/2}-y\right)}}$ \asinh@{iy} = -\tfrac{1}{2}\pi i-\ln@{(y^{2}-1)^{1/2}-y}		arcsinh(I*y) = -(1)/(2)*Pi*I - ln(((y)^(2)- 1)^(1/2)- y)	ArcSinh[I*y] == -Divide[1,2]*Pi*I - Log[((y)^(2)- 1)^(1/2)- y]	Failure	Failure	Failed [4 / 6] Result: -.7e-9+6.283185308*I Test Values: {y = 1.5, y = -2} Result: .1347500000e-10+2.094395102*I Test Values: {y = -.5, y = -2} Result: .1347500000e-10+4.188790205*I Test Values: {y = .5, y = -2} Result: -.2e-8+6.283185308*I Test Values: {y = 2, y = -2}	Successful [Tested: 1]
4.37.E19	${\displaystyle{\displaystyle\operatorname{arccosh}z=\ln\left(+(z^{2}-1)^{1/2}+z\right)}}$ \acosh@@{z} = \ln@{+(z^{2}-1)^{1/2}+z}		arccosh(z) = ln(+((z)^(2)- 1)^(1/2)+ z)	ArcCosh[z] == Log[+((z)^(2)- 1)^(1/2)+ z]	Failure	Failure	Failed [2 / 7] Result: 1.662885893+3.891061519*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 3/2} Result: 1.316957897-4.712388980*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 3/2}	Successful [Tested: 1]
4.37.E19	${\displaystyle{\displaystyle\operatorname{arccosh}z=\ln\left(-(z^{2}-1)^{1/2}+z\right)}}$ \acosh@@{z} = \ln@{-(z^{2}-1)^{1/2}+z}		arccosh(z) = ln(-((z)^(2)- 1)^(1/2)+ z)	ArcCosh[z] == Log[-((z)^(2)- 1)^(1/2)+ z]	Failure	Failure	Failed [5 / 7] Result: 1.316957897+1.570796326*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 3/2} Result: 1.662885893-2.392123788*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 3/2} Result: 1.924847301 Test Values: {z = 1.5, z = 3/2} Result: -.1347500000e-10+2.094395102*I Test Values: {z = .5, z = 3/2} ... skip entries to safe data	Failed [1 / 1] Result: 1.9248473002384139 Test Values: {Rule[z, Rational[3, 2]]}
4.37.E20	${\displaystyle{\displaystyle\operatorname{arccosh}\left(\mathrm{i}y\right)=+\tfrac{1}{2}\pi\mathrm{i}+\ln\left((y^{2}+1)^{1/2}+y\right)}}$ \acosh@{\iunit y} = +\tfrac{1}{2}\pi\iunit+\ln@{(y^{2}+1)^{1/2}+ y}		arccosh(I*y) = +(1)/(2)*Pi*I + ln(((y)^(2)+ 1)^(1/2)+ y)	ArcCosh[I*y] == +Divide[1,2]*Pi*I + Log[((y)^(2)+ 1)^(1/2)+ y]	Failure	Failure	Failed [3 / 6] Result: 2.389526433-3.141592654*I Test Values: {y = -1.5, y = 1/2} Result: .9624236498-3.141592654*I Test Values: {y = -.5, y = 1/2} Result: 2.887270952-3.141592654*I Test Values: {y = -2, y = 1/2}	Successful [Tested: 1]
4.37.E20	${\displaystyle{\displaystyle\operatorname{arccosh}\left(\mathrm{i}y\right)=-\tfrac{1}{2}\pi\mathrm{i}+\ln\left((y^{2}+1)^{1/2}-y\right)}}$ \acosh@{\iunit y} = -\tfrac{1}{2}\pi\iunit+\ln@{(y^{2}+1)^{1/2}- y}		arccosh(I*y) = -(1)/(2)*Pi*I + ln(((y)^(2)+ 1)^(1/2)- y)	ArcCosh[I*y] == -Divide[1,2]*Pi*I + Log[((y)^(2)+ 1)^(1/2)- y]	Failure	Failure	Failed [3 / 6] Result: 2.389526433+3.141592654*I Test Values: {y = 1.5, y = 1/2} Result: .9624236498+3.141592654*I Test Values: {y = .5, y = 1/2} Result: 2.887270952+3.141592654*I Test Values: {y = 2, y = 1/2}	Failed [1 / 1] Result: Complex[0.9624236501192068, 3.141592653589793] Test Values: {Rule[y, Rational[1, 2]]}
4.37.E21	${\displaystyle{\displaystyle\operatorname{arccosh}z=2\ln\left(\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac{z-1}{2}\right)^{1/2}\right)}}$ \acosh@@{z} = 2\ln@{\left(\frac{z+1}{2}\right)^{1/2}+\left(\frac{z-1}{2}\right)^{1/2}}		arccosh(z) = 2*ln(((z + 1)/(2))^(1/2)+((z - 1)/(2))^(1/2))	ArcCosh[z] == 2*Log[(Divide[z + 1,2])^(1/2)+(Divide[z - 1,2])^(1/2)]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 1]
4.37.E22	${\displaystyle{\displaystyle\operatorname{arccosh}x=+\ln\left(i(1-x^{2})^{1/2}+x\right)}}$ \acosh@@{x} = +\ln@{i(1-x^{2})^{1/2}+x}		arccosh(x) = + ln(I*(1 - (x)^(2))^(1/2)+ x)	ArcCosh[x] == + Log[I*(1 - (x)^(2))^(1/2)+ x]	Failure	Failure	Failed [2 / 3] Result: 1.924847301 Test Values: {x = 1.5, x = 1/2} Result: 2.633915796 Test Values: {x = 2, x = 1/2}	Successful [Tested: 1]
4.37.E22	${\displaystyle{\displaystyle\operatorname{arccosh}x=-\ln\left(i(1-x^{2})^{1/2}+x\right)}}$ \acosh@@{x} = -\ln@{i(1-x^{2})^{1/2}+x}		arccosh(x) = - ln(I*(1 - (x)^(2))^(1/2)+ x)	ArcCosh[x] == - Log[I*(1 - (x)^(2))^(1/2)+ x]	Failure	Failure	Failed [1 / 3] Result: .1347500000e-10+2.094395102*I Test Values: {x = .5, x = 1/2}	Failed [1 / 1] Result: Complex[0.0, 2.0943951023931953] Test Values: {Rule[x, Rational[1, 2]]}
4.37.E23	${\displaystyle{\displaystyle\operatorname{arccosh}x=+\pi i+\ln\left((x^{2}-1)^{1/2}-x\right)}}$ \acosh@@{x} = +\pi i+\ln@{(x^{2}-1)^{1/2}-x}		arccosh(x) = + Pi*I + ln(((x)^(2)- 1)^(1/2)- x)	ArcCosh[x] == + Pi*I + Log[((x)^(2)- 1)^(1/2)- x]	Failure	Failure	Failed [3 / 3] Result: 1.924847301-6.283185308*I Test Values: {x = 1.5, x = -2} Result: -.1347500000e-10-4.188790205*I Test Values: {x = .5, x = -2} Result: 2.633915796-6.283185308*I Test Values: {x = 2, x = -2}	Successful [Tested: 1]
4.37.E23	${\displaystyle{\displaystyle\operatorname{arccosh}x=-\pi i+\ln\left((x^{2}-1)^{1/2}-x\right)}}$ \acosh@@{x} = -\pi i+\ln@{(x^{2}-1)^{1/2}-x}		arccosh(x) = - Pi*I + ln(((x)^(2)- 1)^(1/2)- x)	ArcCosh[x] == - Pi*I + Log[((x)^(2)- 1)^(1/2)- x]	Failure	Failure	Failed [3 / 3] Result: 1.924847301+0.*I Test Values: {x = 1.5, x = -2} Result: -.1347500000e-10+2.094395103*I Test Values: {x = .5, x = -2} Result: 2.633915796+0.*I Test Values: {x = 2, x = -2}	Failed [1 / 1] Result: Complex[0.0, 6.283185307179586] Test Values: {Rule[x, -2]}
4.37.E24	${\displaystyle{\displaystyle\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right)}}$ \atanh@@{z} = \tfrac{1}{2}\ln@{\frac{1+z}{1-z}}		arctanh(z) = (1)/(2)*ln((1 + z)/(1 - z))	ArcTanh[z] == Divide[1,2]*Log[Divide[1 + z,1 - z]]	Failure	Failure	Failed [2 / 7] Result: .2e-9-3.141592654*I Test Values: {z = 1.5, z = 1/2} Result: -.2e-9-3.141592654*I Test Values: {z = 2, z = 1/2}	Successful [Tested: 1]
4.37.E25	${\displaystyle{\displaystyle\operatorname{arctanh}x=+\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln\left(\frac{x+1}{x-1}\right)}}$ \atanh@@{x} = +\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln@{\frac{x+1}{x-1}}		arctanh(x) = +(1)/(2)*Pi*I +(1)/(2)*ln((x + 1)/(x - 1))	ArcTanh[x] == +Divide[1,2]*Pi*I +Divide[1,2]*Log[Divide[x + 1,x - 1]]	Failure	Failure	Failed [3 / 3] Result: .2e-9-3.141592654*I Test Values: {x = 1.5, x = -3/2} Result: -.2e-9-3.141592654*I Test Values: {x = .5, x = -3/2} Result: -.2e-9-3.141592654*I Test Values: {x = 2, x = -3/2}	Successful [Tested: 1]
4.37.E25	${\displaystyle{\displaystyle\operatorname{arctanh}x=-\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln\left(\frac{x+1}{x-1}\right)}}$ \atanh@@{x} = -\tfrac{1}{2}\pi i+\tfrac{1}{2}\ln@{\frac{x+1}{x-1}}		arctanh(x) = -(1)/(2)*Pi*I +(1)/(2)*ln((x + 1)/(x - 1))	ArcTanh[x] == -Divide[1,2]*Pi*I +Divide[1,2]*Log[Divide[x + 1,x - 1]]	Failure	Failure	Successful [Tested: 3]	Failed [1 / 1] Result: Complex[-1.1102230246251565*^-16, 3.141592653589793] Test Values: {Rule[x, Rational[-3, 2]]}
4.37.E26	${\displaystyle{\displaystyle z=\sinh w}}$ z = \sinh@@{w}		z = sinh(w)	z == Sinh[w]	Failure	Failure	Failed [70 / 70] Result: .73886869e-2-.1707313589*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -1.358636717+.1952940451*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: -.3586367171-1.536756763*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: -1.724662121-1.170731359*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [70 / 70] Result: Complex[0.007388686967293889, -0.17073135880721174] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.3586367168171445, 0.19529404497722702] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.37.E27	${\displaystyle{\displaystyle z=\cosh w}}$ z = \cosh@@{w}		z = cosh(w)	z == Cosh[w]	Failure	Failure	Failed [70 / 70] Result: -.3617401130+.309246236e-1*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -1.727765517+.3969500276*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: -.7277655170-1.335100780*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: -2.093790921-.9690753764*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [70 / 70] Result: Complex[-0.3617401130796717, 0.030924623731496126] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.7277655168641102, 0.3969500275159349] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.37.E28	${\displaystyle{\displaystyle z=\tanh w}}$ z = \tanh@@{w}		z = tanh(w)	z == Tanh[w]	Failure	Failure	Failed [70 / 70] Result: .736226475e-1+.2564398629*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I} Result: -1.292402756+.6224652669*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)} Result: -.2924027565-1.109585541*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)} Result: -1.658428160-.7435601371*I Test Values: {w = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [70 / 70] Result: Complex[0.07362264736640245, 0.25643986284286624] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.292402756418036, 0.622465266627305] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.37.E29	${\displaystyle{\displaystyle w=\operatorname{Arcsinh}z}}$ w = \Asinh@@{z}		Error	w == ArcSinh[z]	Missing Macro Error	Failure	-	Failed [70 / 70] Result: Complex[0.03458245825512818, 0.12526556729125993] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.524504352246847, -0.28539816339744856] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.37.E29	${\displaystyle{\displaystyle\operatorname{Arcsinh}z=(-1)^{k}\operatorname{arcsinh}z+k\pi i}}$ \Asinh@@{z} = (-1)^{k}\asinh@@{z}+k\pi i		Error	ArcSinh[z] == (- 1)^(k)* ArcSinh[z]+ k*Pi*I	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[1.662885891058621, -2.392123788172313] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.37.E30	${\displaystyle{\displaystyle w=\operatorname{Arccosh}z}}$ w = \Acosh@@{z}		Error	w == ArcCosh[z]	Missing Macro Error	Failure	-	Failed [70 / 70] Result: Complex[0.20754645532203042, -0.28539816339744833] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.03458245825512796, -1.4455307595036366] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.37.E30	${\displaystyle{\displaystyle\operatorname{Arccosh}z=+\operatorname{arccosh}z+2k\pi i}}$ \Acosh@@{z} = +\acosh@@{z}+2k\pi i		Error	ArcCosh[z] == + ArcCosh[z]+ 2*k*Pi*I	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0, -12.566370614359172] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.37.E30	${\displaystyle{\displaystyle\operatorname{Arccosh}z=-\operatorname{arccosh}z+2k\pi i}}$ \Acosh@@{z} = -\acosh@@{z}+2k\pi i		Error	ArcCosh[z] == - ArcCosh[z]+ 2*k*Pi*I	Missing Macro Error	Failure	-	Failed [21 / 21] Result: Complex[1.3169578969248166, -4.71238898038469] Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.3169578969248166, -10.995574287564276] Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.37.E31	${\displaystyle{\displaystyle w=\operatorname{Arctanh}z}}$ w = \Atanh@@{z}		Error	w == ArcTanh[z]	Missing Macro Error	Failure	-	Failed [10 / 10] Result: Complex[0.3167192594503839, 0.49999999999999994] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Rational[1, 2]]} Result: Complex[-1.0493061443340546, 0.8660254037844387] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Rational[1, 2]]} ... skip entries to safe data
4.37.E31	${\displaystyle{\displaystyle\operatorname{Arctanh}z=\operatorname{arctanh}z+k\pi i}}$ \Atanh@@{z} = \atanh@@{z}+k\pi i		Error	ArcTanh[z] == ArcTanh[z]+ k*Pi*I	Missing Macro Error	Failure	-	Failed [3 / 3] Result: Complex[0.0, -3.141592653589793] Test Values: {Rule[k, 1], Rule[z, Rational[1, 2]]} Result: Complex[0.0, -6.283185307179586] Test Values: {Rule[k, 2], Rule[z, Rational[1, 2]]} ... skip entries to safe data
4.38.E4	${\displaystyle{\displaystyle\operatorname{arccosh}z=(2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}\right)}}}$ \acosh@@{z} = (2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}\right)}	${\displaystyle{\displaystyle\Re z>0,|z-1|\leq 2}}$	arccosh(z) = (2*(z - 1))^(1/2)*(1 + sum((- 1)^(n)*(1 * 3 * 5*(2*n - 1))/((2)^(2*n)* factorial(n)*(2*n + 1))*(z - 1)^(n), n = 1..infinity))	ArcCosh[z] == (2*(z - 1))^(1/2)*(1 + Sum[(- 1)^(n)*Divide[1 * 3 * 5*(2*n - 1),(2)^(2*n)* (n)!*(2*n + 1)]*(z - 1)^(n), {n, 1, Infinity}, GenerateConditions->None])	Failure	Failure	Failed [5 / 5] Result: -.5552108774+.3065228369*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -1.819822265-.3498215011*I Test Values: {z = 1/2-1/2*I*3^(1/2)} Result: .5204832489 Test Values: {z = 1.5} Result: -.651724541*I Test Values: {z = .5} ... skip entries to safe data	Failed [5 / 5] Result: Complex[-0.5552108781095244, 0.30652283644847583] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.8198222655846492, -0.34982149976378074] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.38.E8	${\displaystyle{\displaystyle x^{2}-y^{2}=\tfrac{1}{2}}}$ x^{2}-y^{2} = \tfrac{1}{2}		(x)^(2)- (y)^(2) = (1)/(2)	(x)^(2)- (y)^(2) == Divide[1,2]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.38.E9	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsinh}z=(1+z^{2})^{-1/2}}}$ \deriv{}{z}\asinh@@{z} = (1+z^{2})^{-1/2}		diff(arcsinh(z), z) = (1 + (z)^(2))^(- 1/2)	D[ArcSinh[z], z] == (1 + (z)^(2))^(- 1/2)	Successful	Successful	-	Successful [Tested: 7]
4.38.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccosh}z=+(z^{2}-1)^{-1/2}}}$ \deriv{}{z}\acosh@@{z} = +(z^{2}-1)^{-1/2}		diff(arccosh(z), z) = +((z)^(2)- 1)^(- 1/2)	D[ArcCosh[z], z] == +((z)^(2)- 1)^(- 1/2)	Failure	Failure	Failed [2 / 7] Result: -.3933198932-1.467889825*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: -1.000000000+1.732050808*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.38.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccosh}z=-(z^{2}-1)^{-1/2}}}$ \deriv{}{z}\acosh@@{z} = -(z^{2}-1)^{-1/2}		diff(arccosh(z), z) = -((z)^(2)- 1)^(- 1/2)	D[ArcCosh[z], z] == -((z)^(2)- 1)^(- 1/2)	Failure	Failure	Failed [5 / 7] Result: 1.000000000-1.732050808*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: .3933198932+1.467889825*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: 1.788854382 Test Values: {z = 1.5, z = 1/2} Result: -2.309401076*I Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: Complex[0.0, -2.3094010767585034] Test Values: {Rule[z, Rational[1, 2]]}
4.38.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctanh}z=\frac{1}{1-z^{2}}}}$ \deriv{}{z}\atanh@@{z} = \frac{1}{1-z^{2}}		diff(arctanh(z), z) = (1)/(1 - (z)^(2))	D[ArcTanh[z], z] == Divide[1,1 - (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.38.E12	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsch}z=-\frac{1}{z(1+z^{2})^{1/2}}}}$ \deriv{}{z}\acsch@@{z} = -\frac{1}{z(1+z^{2})^{1/2}}		diff(arccsch(z), z) = -(1)/(z*(1 + (z)^(2))^(1/2))	D[ArcCsch[z], z] == -Divide[1,z*(1 + (z)^(2))^(1/2)]	Failure	Failure	Failed [2 / 7] Result: .6696152420e-9-2.000000000*I Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2} Result: -1.074569932+1.074569932*I Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}	Successful [Tested: 1]
4.38.E12	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsch}z=+\frac{1}{z(1+z^{2})^{1/2}}}}$ \deriv{}{z}\acsch@@{z} = +\frac{1}{z(1+z^{2})^{1/2}}		diff(arccsch(z), z) = +(1)/(z*(1 + (z)^(2))^(1/2))	D[ArcCsch[z], z] == +Divide[1,z*(1 + (z)^(2))^(1/2)]	Failure	Failure	Failed [5 / 7] Result: -1.074569932+1.074569932*I Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: .6696152420e-9-2.000000000*I Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2} Result: -.7396002616 Test Values: {z = 1.5, z = 1/2} Result: -3.577708764 Test Values: {z = .5, z = 1/2} ... skip entries to safe data	Failed [1 / 1] Result: -3.5777087639996634 Test Values: {Rule[z, Rational[1, 2]]}
4.38.E13	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsech}z=-\frac{1}{z(1-z^{2})^{1/2}}}}$ \deriv{}{z}\asech@@{z} = -\frac{1}{z(1-z^{2})^{1/2}}		diff(arcsech(z), z) = -(1)/(z*(1 - (z)^(2))^(1/2))	D[ArcSech[z], z] == -Divide[1,z*(1 - (z)^(2))^(1/2)]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
4.38.E14	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccoth}z=\frac{1}{1-z^{2}}}}$ \deriv{}{z}\acoth@@{z} = \frac{1}{1-z^{2}}		diff(arccoth(z), z) = (1)/(1 - (z)^(2))	D[ArcCoth[z], z] == Divide[1,1 - (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
4.38.E15	${\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arcsinh}v=\operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}+v(1+u^{2})^{1/2}\right)}}$ \Asinh@@{u}+\Asinh@@{v} = \Asinh@{u(1+v^{2})^{1/2}+ v(1+u^{2})^{1/2}}		Error	ArcSinh[u]+ ArcSinh[v] == ArcSinh[u*(1 + (v)^(2))^(1/2)+ v*(1 + (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Failed [1 / 100] Result: Complex[-2.633915793849633, 4.440892098500626*^-16] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
4.38.E15	${\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arcsinh}v=\operatorname{Arcsinh}\left(u(1+v^{2})^{1/2}-v(1+u^{2})^{1/2}\right)}}$ \Asinh@@{u}-\Asinh@@{v} = \Asinh@{u(1+v^{2})^{1/2}- v(1+u^{2})^{1/2}}		Error	ArcSinh[u]- ArcSinh[v] == ArcSinh[u*(1 + (v)^(2))^(1/2)- v*(1 + (u)^(2))^(1/2)]	Missing Macro Error	Failure	-	Successful [Tested: 100]
4.38.E16	${\displaystyle{\displaystyle\operatorname{Arccosh}u+\operatorname{Arccosh}v=\operatorname{Arccosh}\left(uv+((u^{2}-1)(v^{2}-1))^{1/2}\right)}}$ \Acosh@@{u}+\Acosh@@{v} = \Acosh@{uv+((u^{2}-1)(v^{2}-1))^{1/2}}		Error	ArcCosh[u]+ ArcCosh[v] == ArcCosh[u*v +(((u)^(2)- 1)*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [60 / 100] Result: Complex[1.3169578969248166, 1.5707963267948966] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.316957896924817, 1.5707963267948966] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.38.E16	${\displaystyle{\displaystyle\operatorname{Arccosh}u-\operatorname{Arccosh}v=\operatorname{Arccosh}\left(uv-((u^{2}-1)(v^{2}-1))^{1/2}\right)}}$ \Acosh@@{u}-\Acosh@@{v} = \Acosh@{uv-((u^{2}-1)(v^{2}-1))^{1/2}}		Error	ArcCosh[u]- ArcCosh[v] == ArcCosh[u*v -(((u)^(2)- 1)*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [80 / 100] Result: Complex[-1.3169578969248166, -1.5707963267948966] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-1.6628858910586213, -3.8910615190072733] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.38.E17	${\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arctanh}v=\operatorname{Arctanh}\left(\frac{u+v}{1+uv}\right)}}$ \Atanh@@{u}+\Atanh@@{v} = \Atanh@{\frac{u+ v}{1+ uv}}		Error	ArcTanh[u]+ ArcTanh[v] == ArcTanh[Divide[u + v,1 + u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.38.E17	${\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arctanh}v=\operatorname{Arctanh}\left(\frac{u-v}{1-uv}\right)}}$ \Atanh@@{u}-\Atanh@@{v} = \Atanh@{\frac{u- v}{1- uv}}		Error	ArcTanh[u]- ArcTanh[v] == ArcTanh[Divide[u - v,1 - u*v]]	Missing Macro Error	Failure	-	Successful [Tested: 1]
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}u+\operatorname{Arccosh}v=\operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1))^{1/2}\right)}}$ \Asinh@@{u}+\Acosh@@{v} = \Asinh@{uv+((1+u^{2})(v^{2}-1))^{1/2}}		Error	ArcSinh[u]+ ArcCosh[v] == ArcSinh[u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [53 / 100] Result: Complex[1.66288587615746, 3.891061504106112] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.6628858910586204, -2.3921237881723125] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}u-\operatorname{Arccosh}v=\operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1))^{1/2}\right)}}$ \Asinh@@{u}-\Acosh@@{v} = \Asinh@{uv-((1+u^{2})(v^{2}-1))^{1/2}}		Error	ArcSinh[u]- ArcCosh[v] == ArcSinh[u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/2)]	Missing Macro Error	Failure	-	Failed [53 / 100] Result: Complex[1.6628858910586208, -2.392123788172313] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.6628858910586208, 3.8910615190072733] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv+((1+u^{2})(v^{2}-1))^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}+u(v^{2}-1)^{1/2}\right)}}$ \Asinh@{uv+((1+u^{2})(v^{2}-1))^{1/2}} = \Acosh@{v(1+u^{2})^{1/2}+ u(v^{2}-1)^{1/2}}		Error	ArcSinh[u*v +((1 + (u)^(2))*((v)^(2)- 1))^(1/2)] == ArcCosh[v*(1 + (u)^(2))^(1/2)+ u*((v)^(2)- 1)^(1/2)]	Missing Macro Error	Failure	-	Failed [65 / 100] Result: Complex[1.4901161193847656*^-8, -3.141592638688632] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.34592799413380415, -2.320265192212377] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E18	${\displaystyle{\displaystyle\operatorname{Arcsinh}\left(uv-((1+u^{2})(v^{2}-1))^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^{1/2}-u(v^{2}-1)^{1/2}\right)}}$ \Asinh@{uv-((1+u^{2})(v^{2}-1))^{1/2}} = \Acosh@{v(1+u^{2})^{1/2}- u(v^{2}-1)^{1/2}}		Error	ArcSinh[u*v -((1 + (u)^(2))*((v)^(2)- 1))^(1/2)] == ArcCosh[v*(1 + (u)^(2))^(1/2)- u*((v)^(2)- 1)^(1/2)]	Missing Macro Error	Failure	-	Failed [86 / 100] Result: Complex[-3.325771782117242, -1.4989377308349603] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-2.9798437879834374, 0.8213274613774169] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} ... skip entries to safe data
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}u+\operatorname{Arccoth}v=\operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right)}}$ \Atanh@@{u}+\Acoth@@{v} = \Atanh@{\frac{uv+ 1}{v+ u}}		Error	ArcTanh[u]+ ArcCoth[v] == ArcTanh[Divide[u*v + 1,v + u]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}u-\operatorname{Arccoth}v=\operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right)}}$ \Atanh@@{u}-\Acoth@@{v} = \Atanh@{\frac{uv- 1}{v- u}}		Error	ArcTanh[u]- ArcCoth[v] == ArcTanh[Divide[u*v - 1,v - u]]	Missing Macro Error	Failure	-	Failed [1 / 10] Result: Indeterminate Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv+1}{v+u}\right)=\operatorname{Arccoth}\left(\frac{v+u}{uv+1}\right)}}$ \Atanh@{\frac{uv+ 1}{v+ u}} = \Acoth@{\frac{v+ u}{uv+ 1}}		Error	ArcTanh[Divide[u*v + 1,v + u]] == ArcCoth[Divide[v + u,u*v + 1]]	Missing Macro Error	Successful	-	Failed [8 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.38.E19	${\displaystyle{\displaystyle\operatorname{Arctanh}\left(\frac{uv-1}{v-u}\right)=\operatorname{Arccoth}\left(\frac{v-u}{uv-1}\right)}}$ \Atanh@{\frac{uv- 1}{v- u}} = \Acoth@{\frac{v- u}{uv- 1}}		Error	ArcTanh[Divide[u*v - 1,v - u]] == ArcCoth[Divide[v - u,u*v - 1]]	Missing Macro Error	Successful	-	Failed [10 / 100] Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.40.E1	${\displaystyle{\displaystyle\int\sinh x\mathrm{d}x=\cosh x}}$ \int\sinh@@{x}\diff{x} = \cosh@@{x}		int(sinh(x), x) = cosh(x)	Integrate[Sinh[x], x, GenerateConditions->None] == Cosh[x]	Successful	Successful	-	Successful [Tested: 3]
4.40.E2	${\displaystyle{\displaystyle\int\cosh x\mathrm{d}x=\sinh x}}$ \int\cosh@@{x}\diff{x} = \sinh@@{x}		int(cosh(x), x) = sinh(x)	Integrate[Cosh[x], x, GenerateConditions->None] == Sinh[x]	Successful	Successful	-	Successful [Tested: 3]
4.40.E3	${\displaystyle{\displaystyle\int\tanh x\mathrm{d}x=\ln\left(\cosh x\right)}}$ \int\tanh@@{x}\diff{x} = \ln@{\cosh@@{x}}		int(tanh(x), x) = ln(cosh(x))	Integrate[Tanh[x], x, GenerateConditions->None] == Log[Cosh[x]]	Successful	Successful	-	Successful [Tested: 3]
4.40.E4	${\displaystyle{\displaystyle\int\operatorname{csch}x\mathrm{d}x=\ln\left(\tanh\left(\tfrac{1}{2}x\right)\right)}}$ \int\csch@@{x}\diff{x} = \ln@{\tanh@{\tfrac{1}{2}x}}	${\displaystyle{\displaystyle 0<x,x<\infty}}$	int(csch(x), x) = ln(tanh((1)/(2)*x))	Integrate[Csch[x], x, GenerateConditions->None] == Log[Tanh[Divide[1,2]*x]]	Successful	Successful	-	Successful [Tested: 3]
4.40.E5	${\displaystyle{\displaystyle\int\operatorname{sech}x\mathrm{d}x=\operatorname{gd}\left(x\right)}}$ \int\sech@@{x}\diff{x} = \Gudermannian@{x}	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	int(sech(x), x) = arctan(sinh(x))	Integrate[Sech[x], x, GenerateConditions->None] == Gudermannian[x]	Successful	Failure	-	Successful [Tested: 3]
4.40.E6	${\displaystyle{\displaystyle\int\coth x\mathrm{d}x=\ln\left(\sinh x\right)}}$ \int\coth@@{x}\diff{x} = \ln@{\sinh@@{x}}	${\displaystyle{\displaystyle 0<x,x<\infty}}$	int(coth(x), x) = ln(sinh(x))	Integrate[Coth[x], x, GenerateConditions->None] == Log[Sinh[x]]	Successful	Successful	-	Successful [Tested: 3]
4.40.E7	${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-x}\frac{\sin\left(ax\right)}{\sinh x}\mathrm{d}x=\tfrac{1}{2}\pi\coth\left(\tfrac{1}{2}\pi a\right)-\frac{1}{a}}}$ \int_{0}^{\infty}e^{-x}\frac{\sin@{ax}}{\sinh@@{x}}\diff{x} = \tfrac{1}{2}\pi\coth@{\tfrac{1}{2}\pi a}-\frac{1}{a}	${\displaystyle{\displaystyle a\neq 0}}$	int(exp(- x)*(sin(a*x))/(sinh(x)), x = 0..infinity) = (1)/(2)*Pi*coth((1)/(2)*Pi*a)-(1)/(a)	Integrate[Exp[- x]*Divide[Sin[a*x],Sinh[x]], {x, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Pi*Coth[Divide[1,2]*Pi*a]-Divide[1,a]	Failure	Aborted	Successful [Tested: 6]	Successful [Tested: 6]
4.40.E8	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\sinh\left(ax\right)}{\sinh\left(\pi x\right)}\mathrm{d}x=\tfrac{1}{2}\tan\left(\tfrac{1}{2}a\right)}}$ \int_{0}^{\infty}\frac{\sinh@{ax}}{\sinh@{\pi x}}\diff{x} = \tfrac{1}{2}\tan@{\tfrac{1}{2}a}	${\displaystyle{\displaystyle-\pi<a,a<\pi}}$	int((sinh(a*x))/(sinh(Pi*x)), x = 0..infinity) = (1)/(2)*tan((1)/(2)*a)	Integrate[Divide[Sinh[a*x],Sinh[Pi*x]], {x, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Tan[Divide[1,2]*a]	Failure	Aborted	Successful [Tested: 6]	Skipped - Because timed out
4.40.E9	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh\left(\tfrac{1}{2}x\right)\right)^{2}}\mathrm{d}x=\frac{4\pi a}{\sin\left(\pi a\right)}}}$ \int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh@{\tfrac{1}{2}x}\right)^{2}}\diff{x} = \frac{4\pi a}{\sin@{\pi a}}	${\displaystyle{\displaystyle-1<a,a<1}}$	int((exp(a*x))/((cosh((1)/(2)*x))^(2)), x = - infinity..infinity) = (4*Pi*a)/(sin(Pi*a))	Integrate[Divide[Exp[a*x],(Cosh[Divide[1,2]*x])^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == Divide[4*Pi*a,Sin[Pi*a]]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
4.40.E10	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\tanh\left(ax\right)-\tanh\left(bx\right)}{x}\mathrm{d}x=\ln\left(\frac{a}{b}\right)}}$ \int_{0}^{\infty}\frac{\tanh@{ax}-\tanh@{bx}}{x}\diff{x} = \ln@{\frac{a}{b}}	${\displaystyle{\displaystyle a>0,b>0}}$	int((tanh(a*x)- tanh(b*x))/(x), x = 0..infinity) = ln((a)/(b))	Integrate[Divide[Tanh[a*x]- Tanh[b*x],x], {x, 0, Infinity}, GenerateConditions->None] == Log[Divide[a,b]]	Skipped - Unable to analyze test case: Null	Skipped - Unable to analyze test case: Null	-	-
4.40.E11	${\displaystyle{\displaystyle\int\operatorname{arcsinh}x\mathrm{d}x=x\operatorname{arcsinh}x-(1+x^{2})^{1/2}}}$ \int\asinh@@{x}\diff{x} = x\asinh@@{x}-(1+x^{2})^{1/2}		int(arcsinh(x), x) = x*arcsinh(x)-(1 + (x)^(2))^(1/2)	Integrate[ArcSinh[x], x, GenerateConditions->None] == x*ArcSinh[x]-(1 + (x)^(2))^(1/2)	Successful	Successful	-	Successful [Tested: 3]
4.40.E12	${\displaystyle{\displaystyle\int\operatorname{arccosh}x\mathrm{d}x=x\operatorname{arccosh}x-(x^{2}-1)^{1/2}}}$ \int\acosh@@{x}\diff{x} = x\acosh@@{x}-(x^{2}-1)^{1/2}	${\displaystyle{\displaystyle 1<x,x<\infty}}$	int(arccosh(x), x) = x*arccosh(x)-((x)^(2)- 1)^(1/2)	Integrate[ArcCosh[x], x, GenerateConditions->None] == x*ArcCosh[x]-((x)^(2)- 1)^(1/2)	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
4.40.E13	${\displaystyle{\displaystyle\int\operatorname{arctanh}x\mathrm{d}x=x\operatorname{arctanh}x+\tfrac{1}{2}\ln\left(1-x^{2}\right)}}$ \int\atanh@@{x}\diff{x} = x\atanh@@{x}+\tfrac{1}{2}\ln@{1-x^{2}}	${\displaystyle{\displaystyle-1<x,x<1}}$	int(arctanh(x), x) = x*arctanh(x)+(1)/(2)*ln(1 - (x)^(2))	Integrate[ArcTanh[x], x, GenerateConditions->None] == x*ArcTanh[x]+Divide[1,2]*Log[1 - (x)^(2)]	Successful	Successful	-	Successful [Tested: 1]
4.40.E14	${\displaystyle{\displaystyle\int\operatorname{arccsch}x\mathrm{d}x=x\operatorname{arccsch}x+\operatorname{arcsinh}x}}$ \int\acsch@@{x}\diff{x} = x\acsch@@{x}+\asinh@@{x}	${\displaystyle{\displaystyle 0<x,x<\infty}}$	int(arccsch(x), x) = x*arccsch(x)+ arcsinh(x)	Integrate[ArcCsch[x], x, GenerateConditions->None] == x*ArcCsch[x]+ ArcSinh[x]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
4.40.E15	${\displaystyle{\displaystyle\int\operatorname{arcsech}x\mathrm{d}x=x\operatorname{arcsech}x+\operatorname{arcsin}x}}$ \int\asech@@{x}\diff{x} = x\asech@@{x}+\asin@@{x}	${\displaystyle{\displaystyle 0<x,x<1}}$	int(arcsech(x), x) = x*arcsech(x)+ arcsin(x)	Integrate[ArcSech[x], x, GenerateConditions->None] == x*ArcSech[x]+ ArcSin[x]	Failure	Successful	Failed [1 / 1] Result: -1.570796327 Test Values: {x = .5}	Successful [Tested: 1]
4.40.E16	${\displaystyle{\displaystyle\int\operatorname{arccoth}x\mathrm{d}x=x\operatorname{arccoth}x+\tfrac{1}{2}\ln\left(x^{2}-1\right)}}$ \int\acoth@@{x}\diff{x} = x\acoth@@{x}+\tfrac{1}{2}\ln@{x^{2}-1}	${\displaystyle{\displaystyle 1<x,x<\infty}}$	int(arccoth(x), x) = x*arccoth(x)+(1)/(2)*ln((x)^(2)- 1)	Integrate[ArcCoth[x], x, GenerateConditions->None] == x*ArcCoth[x]+Divide[1,2]*Log[(x)^(2)- 1]	Successful	Failure	-	Failed [1 / 1] Result: Complex[0.0, -1.5707963267948966] Test Values: {Rule[x, Rational[1, 2]]}
4.42.E1	${\displaystyle{\displaystyle\sin A=\frac{a}{c}}}$ \sin@@{A} = \frac{a}{c}		sin(A) = (a)/(c)	Sin[A] == Divide[a,c]	Failure	Failure	Failed [300 / 300] Result: -.1410196655+.3375964631*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -1.5} Result: 1.858980334+.3375964631*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = 1.5} Result: -2.141019666+.3375964631*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -.5} Result: 3.858980334+.3375964631*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.14101966569986213, 0.33759646322287] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5]} Result: Complex[1.8589803343001379, 0.33759646322287] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, 1.5]} ... skip entries to safe data
4.42.E1	${\displaystyle{\displaystyle\frac{a}{c}=\frac{1}{\csc A}}}$ \frac{a}{c} = \frac{1}{\csc@@{A}}		(a)/(c) = (1)/(csc(A))	Divide[a,c] == Divide[1,Csc[A]]	Failure	Failure	Failed [300 / 300] Result: .1410196654-.3375964632*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -1.5} Result: -1.858980335-.3375964632*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = 1.5} Result: 2.141019665-.3375964632*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = -.5} Result: -3.858980335-.3375964632*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.14101966569986213, -0.33759646322287] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5]} Result: Complex[-1.8589803343001379, -0.33759646322287] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, 1.5]} ... skip entries to safe data
4.42.E2	${\displaystyle{\displaystyle\cos A=\frac{b}{c}}}$ \cos@@{A} = \frac{b}{c}		cos(A) = (b)/(c)	Cos[A] == Divide[b,c]	Failure	Failure	Failed [300 / 300] Result: -.2694569811-.3969495503*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -1.5} Result: 1.730543019-.3969495503*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5} Result: -2.269456981-.3969495503*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5} Result: 3.730543019-.3969495503*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.2694569809427748, -0.3969495502290325] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5]} Result: Complex[1.730543019057225, -0.3969495502290325] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, 1.5]} ... skip entries to safe data
4.42.E2	${\displaystyle{\displaystyle\frac{b}{c}=\frac{1}{\sec A}}}$ \frac{b}{c} = \frac{1}{\sec@@{A}}		(b)/(c) = (1)/(sec(A))	Divide[b,c] == Divide[1,Sec[A]]	Failure	Failure	Failed [300 / 300] Result: .2694569810+.3969495505*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -1.5} Result: -1.730543019+.3969495505*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5} Result: 2.269456981+.3969495505*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5} Result: -3.730543019+.3969495505*I Test Values: {A = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.2694569809427748, 0.3969495502290325] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5]} Result: Complex[-1.730543019057225, 0.3969495502290325] Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, 1.5]} ... skip entries to safe data
4.42.E3	${\displaystyle{\displaystyle\tan A=\frac{a}{b}}}$ \tan@@{A} = \frac{a}{b}		tan(A) = (a)/(b)	Tan[A] == Divide[a,b]	Failure	Failure	Failed [300 / 300] Result: -.2860691196+.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5} Result: 1.713930880+.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5} Result: -2.286069120+.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5} Result: 3.713930880+.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.2860691197539781, 0.8500402971922751] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5]} Result: Complex[1.7139308802460218, 0.8500402971922751] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, 1.5]} ... skip entries to safe data
4.42.E3	${\displaystyle{\displaystyle\frac{a}{b}=\frac{1}{\cot A}}}$ \frac{a}{b} = \frac{1}{\cot@@{A}}		(a)/(b) = (1)/(cot(A))	Divide[a,b] == Divide[1,Cot[A]]	Failure	Failure	Failed [300 / 300] Result: .2860691196-.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5} Result: -1.713930880-.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5} Result: 2.286069120-.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5} Result: -3.713930880-.8500402975*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.2860691197539781, -0.8500402971922751] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5]} Result: Complex[-1.7139308802460218, -0.8500402971922751] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, 1.5]} ... skip entries to safe data
4.42.E4	${\displaystyle{\displaystyle\frac{a}{\sin A}=\frac{b}{\sin B}}}$ \frac{a}{\sin@@{A}} = \frac{b}{\sin@@{B}}		(a)/(sin(A)) = (b)/(sin(B))	Divide[a,Sin[A]] == Divide[b,Sin[B]]	Failure	Failure	Failed [288 / 300] Result: -3.025222791+1.188973104*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5} Result: -1.008407597+.3963243680*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5} Result: -2.016815194+.7926487360*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5} Result: .5042037985-.1981621840*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -2} ... skip entries to safe data	Failed [290 / 300] Result: Complex[-2.3601096690692955, -0.4904383214455733] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.6651131226742772, 1.6794114261511237] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.42.E4	${\displaystyle{\displaystyle\frac{b}{\sin B}=\frac{c}{\sin C}}}$ \frac{b}{\sin@@{B}} = \frac{c}{\sin@@{C}}		(b)/(sin(B)) = (c)/(sin(C))	Divide[b,Sin[B]] == Divide[c,Sin[C]]	Failure	Failure	Failed [288 / 300] Result: -3.025222791+1.188973104*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5} Result: -1.008407597+.3963243680*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5} Result: -2.016815194+.7926487360*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5} Result: .5042037985-.1981621840*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -2} ... skip entries to safe data	Failed [290 / 300] Result: Complex[-2.3601096690692955, -0.4904383214455733] Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.6651131226742772, 1.6794114261511237] Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.42.E5	${\displaystyle{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C}}$ c^{2} = a^{2}+b^{2}-2ab\cos@@{C}		(c)^(2) = (a)^(2)+ (b)^(2)- 2*a*b*cos(C)	(c)^(2) == (a)^(2)+ (b)^(2)- 2*a*b*Cos[C]	Failure	Failure	Failed [300 / 300] Result: 1.037443585-1.786272976*I Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5} Result: 1.037443585-1.786272976*I Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5} Result: -.962556415-1.786272976*I Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5} Result: -.962556415-1.786272976*I Test Values: {C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.0374435857575133, -1.7862729760306462] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[3.274944825888497, 2.1108391932082666] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.42.E6	${\displaystyle{\displaystyle a=b\cos C+c\cos B}}$ a = b\cos@@{C}+c\cos@@{B}		a = b*cos(C)+ c*cos(B)	a == b*Cos[C]+ c*Cos[B]	Failure	Failure	Failed [300 / 300] Result: .691629057-1.190848651*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5} Result: -1.5 Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5} Result: -.38913962e-1-.7938991006*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5} Result: -.7694569811-.3969495503*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.6916290571716757, -1.1908486506870974] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.4374628038820034, 0.10818873905920678] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.42.E7	${\displaystyle{\displaystyle\hbox{area}=\tfrac{1}{2}bc\sin A}}$ \hbox{area} = \tfrac{1}{2}bc\sin@@{A}		a*r*exp(1)*a = (1)/(2)*b*c*sin(A)	a*r*E*a == Divide[1,2]*b*c*Sin[A]	Failure	Failure	Failed [300 / 300] Result: -10.14055405-.3797960210*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = -1.5} Result: 8.207848294-.3797960210*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = 1.5} Result: -4.024419932-.3797960210*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = -.5} Result: 2.091714180-.3797960210*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, r = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-10.140554047136932, -0.37979602112572874] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[r, -1.5]} Result: Complex[8.207848294961623, -0.37979602112572874] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[r, 1.5]} ... skip entries to safe data
4.42.E7	${\displaystyle{\displaystyle\tfrac{1}{2}bc\sin A=\left(s(s-a)(s-b)(s-c)\right)^{1/2}}}$ \tfrac{1}{2}bc\sin@@{A} = \left(s(s-a)(s-b)(s-c)\right)^{1/2}		(1)/(2)*b*c*sin(A) = (s*(s - a)*(s - b)*(s - c))^(1/2)	Divide[1,2]*b*c*Sin[A] == (s*(s - a)*(s - b)*(s - c))^(1/2)	Failure	Failure	Failed [300 / 300] Result: .9663528763+.3797960210*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = -1.5} Result: -5.397608155+.3797960210*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = 1.5} Result: .9663528763-.3273107602*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = -.5} Result: -1.033647124+.3797960210*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5, s = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.9663528760876551, 0.37979602112572874] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[s, -1.5]} Result: Complex[-5.397608154591272, 0.37979602112572874] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5], Rule[s, 1.5]} ... skip entries to safe data
4.42.E8	${\displaystyle{\displaystyle\cos a=\cos b\cos c+\sin b\sin c\cos A}}$ \cos@@{a} = \cos@@{b}\cos@@{c}+\sin@@{b}\sin@@{c}\cos@@{A}		cos(a) = cos(b)*cos(c)+ sin(b)*sin(c)*cos(A)	Cos[a] == Cos[b]*Cos[c]+ Sin[b]*Sin[c]*Cos[A]	Failure	Failure	Failed [300 / 300] Result: -.6611541130+.3949633133*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5} Result: .7926210130-.3949633133*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5} Result: -.3407041550+.1898310285*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5} Result: .3580230890-.1898310285*I Test Values: {A = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.6611541132159315, 0.3949633132423481] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, -1.5]} Result: Complex[0.7926210131517828, -0.3949633132423481] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[c, 1.5]} ... skip entries to safe data
4.42.E9	${\displaystyle{\displaystyle\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}}}$ \frac{\sin@@{A}}{\sin@@{a}} = \frac{\sin@@{B}}{\sin@@{b}}		(sin(A))/(sin(a)) = (sin(B))/(sin(b))	Divide[Sin[A],Sin[a]] == Divide[Sin[B],Sin[b]]	Failure	Failure	Failed [288 / 300] Result: -1.722274990-.6768885409*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5} Result: .9305491492+.3657244397*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5} Result: -2.652824140-1.042612981*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5} Result: .835262737e-1+.328274973e-1*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -2} ... skip entries to safe data	Failed [286 / 300] Result: Complex[-1.5335532645785146, 0.5223487441958409] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.18872172594452297, -1.1992372855051223] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.42.E9	${\displaystyle{\displaystyle\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}}}$ \frac{\sin@@{B}}{\sin@@{b}} = \frac{\sin@@{C}}{\sin@@{c}}		(sin(B))/(sin(b)) = (sin(C))/(sin(c))	Divide[Sin[B],Sin[b]] == Divide[Sin[C],Sin[c]]	Failure	Failure	Failed [288 / 300] Result: -1.722274990-.6768885409*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = 1.5} Result: .9305491492+.3657244397*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -.5} Result: -2.652824140-1.042612981*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = .5} Result: .835262737e-1+.328274973e-1*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, b = -1.5, c = -2} ... skip entries to safe data	Failed [286 / 300] Result: Complex[-1.5335532645785146, 0.5223487441958409] Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-0.18872172594452297, -1.1992372855051223] Test Values: {Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5], Rule[C, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]} ... skip entries to safe data
4.42.E10	${\displaystyle{\displaystyle\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A}}$ \sin@@{a}\cos@@{B} = \cos@@{b}\sin@@{c}-\sin@@{b}\cos@@{c}\cos@@{A}		sin(a)*cos(B) = cos(b)*sin(c)- sin(b)*cos(c)*cos(A)	Sin[a]*Cos[B] == Cos[b]*Sin[c]- Sin[b]*Cos[c]*Cos[A]	Failure	Failure	Failed [300 / 300] Result: -.7097001135+.4239639484*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -1.5} Result: -.8508201215+.4239639484*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = 1.5} Result: -1.334305598+.7434385530*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = -.5} Result: -1.402132040+.7434385530*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5, c = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.7097001133469564, 0.4239639481520351] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, -1.5]} Result: Complex[-0.8508201214068235, 0.4239639481520351] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[c, 1.5]} ... skip entries to safe data
4.42.E11	${\displaystyle{\displaystyle\cos a\cos C=\sin a\cot b-\sin C\cot B}}$ \cos@@{a}\cos@@{C} = \sin@@{a}\cot@@{b}-\sin@@{C}\cot@@{B}		cos(a)*cos(C) = sin(a)*cot(b)- sin(C)*cot(B)	Cos[a]*Cos[C] == Sin[a]*Cot[b]- Sin[C]*Cot[B]	Failure	Failure	Failed [300 / 300] Result: .7114823860-.4250286508*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -1.5} Result: .8529567893-.4250286508*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = 1.5} Result: -1.043682738-.4250286508*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = -.5} Result: 2.608121914-.4250286508*I Test Values: {B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5, b = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.7114823862555057, -0.42502865061548756] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.21981752916457492, 0.9933277802647092] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.42.E12	${\displaystyle{\displaystyle\cos A=-\cos B\cos C+\sin B\sin C\cos a}}$ \cos@@{A} = -\cos@@{B}\cos@@{C}+\sin@@{B}\sin@@{C}\cos@@{a}		cos(A) = - cos(B)*cos(C)+ sin(B)*sin(C)*cos(a)	Cos[A] == - Cos[B]*Cos[C]+ Sin[B]*Sin[C]*Cos[a]	Failure	Failure	Failed [300 / 300] Result: 1.062535945-1.017952978*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -1.5} Result: 1.062535945-1.017952978*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = 1.5} Result: .5591646152-1.485905089*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = -.5} Result: .5591646152-1.485905089*I Test Values: {A = 1/2*3^(1/2)+1/2*I, B = 1/2*3^(1/2)+1/2*I, C = 1/2*3^(1/2)+1/2*I, a = .5} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.0625359450203713, -1.017952977441946] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.8749374794081675, -0.5777856599721184] Test Values: {Rule[a, -1.5], Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.43#Ex1	${\displaystyle{\displaystyle A=\left(-\tfrac{4}{3}p\right)^{1/2}}}$ A = \left(-\tfrac{4}{3}p\right)^{1/2}		A = (-(4)/(3)*p)^(1/2)	A == (-Divide[4,3]*p)^(1/2)	Skipped - no semantic math	Skipped - no semantic math	-	-
4.43#Ex2	${\displaystyle{\displaystyle B=\left(\tfrac{4}{3}p\right)^{1/2}}}$ B = \left(\tfrac{4}{3}p\right)^{1/2}		B = ((4)/(3)*p)^(1/2)	B == (Divide[4,3]*p)^(1/2)	Skipped - no semantic math	Skipped - no semantic math	-	-
4.43.E2	${\displaystyle{\displaystyle z^{3}+pz+q=0}}$ z^{3}+pz+q = 0		(z)^(3)+ p*z + q = 0	(z)^(3)+ p*z + q == 0	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45.E1	${\displaystyle{\displaystyle y=x^{2^{-m}}-1}}$ y = x^{2^{-m}}-1		y = (x)^((2)^(- m))- 1	y == (x)^((2)^(- m))- 1	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45.E2	${\displaystyle{\displaystyle\ln x=2^{m}\ln\left(1+y\right)}}$ \ln@@{x} = 2^{m}\ln@{1+y}		ln(x) = (2)^(m)* ln(1 + y)	Log[x] == (2)^(m)* Log[1 + y]	Failure	Failure	Failed [54 / 54] Result: 1.791759469-6.283185308*I Test Values: {x = 1.5, y = -1.5, m = 1} Result: 3.178053830-12.56637062*I Test Values: {x = 1.5, y = -1.5, m = 2} Result: 5.950642553-25.13274123*I Test Values: {x = 1.5, y = -1.5, m = 3} Result: -1.427116356 Test Values: {x = 1.5, y = 1.5, m = 1} ... skip entries to safe data	Failed [54 / 54] Result: Complex[1.791759469228055, -6.283185307179586] Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[y, -1.5]} Result: Complex[3.1780538303479453, -12.566370614359172] Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[y, -1.5]} ... skip entries to safe data
4.45.E3	${\displaystyle{\displaystyle\ln x=\ln\xi+m\ln 10}}$ \ln@@{x} = \ln@@{\xi}+m\ln@@{10}		ln(x) = ln(xi)+ m*ln(10)	Log[x] == Log[\[Xi]]+ m*Log[10]	Failure	Failure	Failed [90 / 90] Result: -1.897119985-.5235987755*I Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I, m = 1} Result: -4.199705078-.5235987755*I Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I, m = 2} Result: -6.502290171-.5235987755*I Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I, m = 3} Result: -1.897119985-2.094395102*I Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2), m = 1} ... skip entries to safe data	Failed [90 / 90] Result: Complex[-1.8971199848858815, -0.5235987755982988] Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-4.199705077879927, -0.5235987755982988] Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.45#Ex1	${\displaystyle{\displaystyle m=\left\lfloor\frac{x}{\ln 10}+\frac{1}{2}\right\rfloor}}$ m = \floor{\frac{x}{\ln@@{10}}+\frac{1}{2}}		m = floor((x)/(ln(10))+(1)/(2))	m == Floor[Divide[x,Log[10]]+Divide[1,2]]	Failure	Failure	Failed [7 / 9] Result: 1. Test Values: {x = 1.5, m = 2} Result: 2. Test Values: {x = 1.5, m = 3} Result: 1. Test Values: {x = .5, m = 1} Result: 2. Test Values: {x = .5, m = 2} ... skip entries to safe data	Failed [7 / 9] Result: 1.0 Test Values: {Rule[m, 2], Rule[x, 1.5]} Result: 2.0 Test Values: {Rule[m, 3], Rule[x, 1.5]} ... skip entries to safe data
4.45#Ex2	${\displaystyle{\displaystyle y=x-m\ln 10}}$ y = x-m\ln@@{10}		y = x - m*ln(10)	y == x - m*Log[10]	Failure	Failure	Failed [54 / 54] Result: -.697414907 Test Values: {x = 1.5, y = -1.5, m = 1} Result: 1.605170186 Test Values: {x = 1.5, y = -1.5, m = 2} Result: 3.907755279 Test Values: {x = 1.5, y = -1.5, m = 3} Result: 2.302585093 Test Values: {x = 1.5, y = 1.5, m = 1} ... skip entries to safe data	Failed [54 / 54] Result: -0.6974149070059541 Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[y, -1.5]} Result: 1.6051701859880918 Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[y, -1.5]} ... skip entries to safe data
4.45.E5	${\displaystyle{\displaystyle e^{x}=10^{m}e^{y}}}$ e^{x} = 10^{m}e^{y}		exp(x) = (10)^(m)* exp(y)	Exp[x] == (10)^(m)* Exp[y]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45#Ex3	${\displaystyle{\displaystyle m=\left\lfloor\xi+\tfrac{1}{2}\right\rfloor}}$ m = \floor{\xi+\tfrac{1}{2}}		m = floor(xi +(1)/(2))	m == Floor[\[Xi]+Divide[1,2]]	Failure	Failure	Failed [26 / 30] Result: 1. Test Values: {xi = 1/2*3^(1/2)+1/2*I, m = 2} Result: 2. Test Values: {xi = 1/2*3^(1/2)+1/2*I, m = 3} Result: 1. Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 1} Result: 2. Test Values: {xi = -1/2+1/2*I*3^(1/2), m = 2} ... skip entries to safe data	Failed [26 / 30] Result: 1.0 Test Values: {Rule[m, 2], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: 2.0 Test Values: {Rule[m, 3], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.45#Ex4	${\displaystyle{\displaystyle\theta=\pi(\xi-m)}}$ \theta = \pi(\xi-m)		theta = Pi*(xi - m)	\[Theta] == Pi*(\[Xi]- m)	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45#Ex5	${\displaystyle{\displaystyle\sin x=(-1)^{m}\sin\theta}}$ \sin@@{x} = (-1)^{m}\sin@@{\theta}		sin(x) = (- 1)^(m)* sin(theta)	Sin[x] == (- 1)^(m)* Sin[\[Theta]]	Failure	Failure	Failed [81 / 90] Result: 1.856475321+.3375964631*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 1} Result: .1385146521-.3375964631*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 2} Result: 1.856475321+.3375964631*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 3} Result: 1.338405873+.3375964631*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = .5, m = 1} ... skip entries to safe data	Failed [81 / 90] Result: Complex[1.8564753209041922, 0.33759646322287] Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.13851465230391657, -0.33759646322287] Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.45#Ex6	${\displaystyle{\displaystyle\cos x=(-1)^{m}\cos\theta}}$ \cos@@{x} = (-1)^{m}\cos@@{\theta}		cos(x) = (- 1)^(m)* cos(theta)	Cos[x] == (- 1)^(m)* Cos[\[Theta]]	Failure	Failure	Failed [84 / 90] Result: .8012802206-.3969495503*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 1} Result: -.6598058172+.3969495503*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 2} Result: .8012802206-.3969495503*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 3} Result: 1.608125581-.3969495503*I Test Values: {theta = 1/2*3^(1/2)+1/2*I, x = .5, m = 1} ... skip entries to safe data	Failed [84 / 90] Result: Complex[0.8012802207249281, -0.3969495502290325] Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.6598058173895223, 0.3969495502290325] Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.45.E8	${\displaystyle{\displaystyle 2\operatorname{arctan}\frac{x}{1+(1+x^{2})^{1/2}}=\operatorname{arctan}x}}$ 2\atan@@{\frac{x}{1+(1+x^{2})^{1/2}}} = \atan@@{x}	${\displaystyle{\displaystyle 0<x,x<\infty}}$	2*arctan((x)/(1 +(1 + (x)^(2))^(1/2))) = arctan(x)	2*ArcTan[Divide[x,1 +(1 + (x)^(2))^(1/2)]] == ArcTan[x]	Successful	Failure	-	Successful [Tested: 3]
4.45.E9	${\displaystyle{\displaystyle x_{n}=\frac{x_{n-1}}{1+(1+x^{2}_{n-1})^{1/2}}}}$ x_{n} = \frac{x_{n-1}}{1+(1+x^{2}_{n-1})^{1/2}}		x[n] = (x[n - 1])/(1 +(1 + (x[n - 1])^(2))^(1/2))	Subscript[x, n] == Divide[Subscript[x, n - 1],1 +(1 + (Subscript[x, n - 1])^(2))^(1/2)]	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45.E10	${\displaystyle{\displaystyle\operatorname{arctan}x=2^{n}\operatorname{arctan}x_{n}}}$ \atan@@{x} = 2^{n}\atan@@{x_{n}}		arctan(x) = (2)^(n)* arctan(x[n])	ArcTan[x] == (2)^(n)* ArcTan[Subscript[x, n]]	Failure	Failure	Failed [90 / 90] Result: -.5880026038-.5493061442*I Test Values: {x = 1.5, x[n] = 1/2*3^(1/2)+1/2*I, n = 1} Result: -2.158798931-1.098612288*I Test Values: {x = 1.5, x[n] = 1/2*3^(1/2)+1/2*I, n = 2} Result: -5.300391585-2.197224577*I Test Values: {x = 1.5, x[n] = 1/2*3^(1/2)+1/2*I, n = 3} Result: 2.553590050-1.316957897*I Test Values: {x = 1.5, x[n] = -1/2+1/2*I*3^(1/2), n = 1} ... skip entries to safe data	Failed [90 / 90] Result: Complex[-0.5880026035475677, -0.5493061443340551] Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[x, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.1587989303424644, -1.0986122886681102] Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[x, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
4.45#Ex7	${\displaystyle{\displaystyle x_{1}=0.90000\dots}}$ x_{1} = 0.90000\dots		x[1] = 0.90000	Subscript[x, 1] == 0.90000	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45#Ex8	${\displaystyle{\displaystyle x_{2}=0.38373\dots}}$ x_{2} = 0.38373\dots		x[2] = 0.38373	Subscript[x, 2] == 0.38373	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45#Ex9	${\displaystyle{\displaystyle x_{3}=0.18528\dots}}$ x_{3} = 0.18528\dots		x[3] = 0.18528	Subscript[x, 3] == 0.18528	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45#Ex10	${\displaystyle{\displaystyle x_{4}=0.09185\dots}}$ x_{4} = 0.09185\dots		x[4] = 0.09185	Subscript[x, 4] == 0.09185	Skipped - no semantic math	Skipped - no semantic math	-	-
4.45.E13	${\displaystyle{\displaystyle\operatorname{arctan}x=16\operatorname{arctan}x_{4}}}$ \atan@@{x} = 16\atan@@{x_{4}}		arctan(x) = 16*arctan(x[4])	ArcTan[x] == 16*ArcTan[Subscript[x, 4]]	Failure	Failure	Failed [30 / 30] Result: -11.58357690-4.394449154*I Test Values: {x = 1.5, x[4] = 1/2*3^(1/2)+1/2*I} Result: 13.54916434-10.53566318*I Test Values: {x = 1.5, x[4] = -1/2+1/2*I*3^(1/2)} Result: -11.58357690+10.53566318*I Test Values: {x = 1.5, x[4] = 1/2-1/2*I*3^(1/2)} Result: 13.54916434+4.394449154*I Test Values: {x = 1.5, x[4] = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-11.583576891111845, -4.394449154672441] Test Values: {Rule[x, 1.5], Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[13.549164337606502, -10.535663175398536] Test Values: {Rule[x, 1.5], Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.45.E13	${\displaystyle{\displaystyle 16\operatorname{arctan}x_{4}=1.46563\dots}}$ 16\atan@@{x_{4}} = 1.46563\dots		16*arctan(x[4]) = 1.46563	16*ArcTan[Subscript[x, 4]] == 1.46563	Failure	Failure	Failed [10 / 10] Result: 11.10074062+4.394449154*I Test Values: {x[4] = 1/2*3^(1/2)+1/2*I} Result: -14.03200062+10.53566318*I Test Values: {x[4] = -1/2+1/2*I*3^(1/2)} Result: 11.10074062-10.53566318*I Test Values: {x[4] = 1/2-1/2*I*3^(1/2)} Result: -14.03200062-4.394449154*I Test Values: {x[4] = -1/2*3^(1/2)-1/2*I} ... skip entries to safe data	Failed [10 / 10] Result: Complex[11.100740614359175, 4.394449154672441] Test Values: {Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-14.032000614359173, 10.535663175398536] Test Values: {Rule[Subscript[x, 4], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
4.45.E15	${\displaystyle{\displaystyle\ln z=\ln|z|+i\operatorname{ph}z}}$ \ln@@{z} = \ln@@{|z|}+i\phase@@{z}	${\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z,\operatorname{ph}z\leq\pi}}$	ln(z) = ln(abs(z))+ I*argument(z)	Log[z] == Log[Abs[z]]+ I*Arg[z]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
4.45.E16	${\displaystyle{\displaystyle e^{z}=e^{\Re z}(\cos\left(\Im z\right)+i\sin\left(\Im z\right))}}$ e^{z} = e^{\realpart@@{z}}(\cos@{\imagpart@@{z}}+i\sin@{\imagpart@@{z}})		exp(z) = exp(Re(z))*(cos(Im(z))+ I*sin(Im(z)))	Exp[z] == Exp[Re[z]]*(Cos[Im[z]]+ I*Sin[Im[z]])	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]