4.40

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
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]]}