Elementary Functions - 4.40 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.40.E1 ∫ sinh ⁡ x ⁢ d x = cosh ⁡ x 𝑥 𝑥 𝑥 {\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 ∫ cosh ⁡ x ⁢ d x = sinh ⁡ x 𝑥 𝑥 𝑥 {\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 ∫ tanh ⁡ x ⁢ d x = ln ⁡ ( cosh ⁡ x ) 𝑥 𝑥 𝑥 {\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 ∫ csch ⁡ x ⁢ d x = ln ⁡ ( tanh ⁡ ( 1 2 ⁢ x ) ) 𝑥 𝑥 1 2 𝑥 {\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}}		 0 < x , x < ∞ formulae-sequence 0 𝑥 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 ∫ sech ⁡ x ⁢ d x = gd ⁡ ( x ) 𝑥 𝑥 Gudermannian 𝑥 {\displaystyle{\displaystyle\int\operatorname{sech}x\mathrm{d}x=\operatorname{% gd}\left(x\right)}}
\int\sech@@{x}\diff{x} = \Gudermannian@{x}		 - ∞ < x , x < ∞ formulae-sequence 𝑥 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 ∫ coth ⁡ x ⁢ d x = ln ⁡ ( sinh ⁡ x ) hyperbolic-cotangent 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle\int\coth x\mathrm{d}x=\ln\left(\sinh x\right)}}
\int\coth@@{x}\diff{x} = \ln@{\sinh@@{x}}		 0 < x , x < ∞ formulae-sequence 0 𝑥 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 ∫ 0 ∞ e - x ⁢ sin ⁡ ( a ⁢ x ) sinh ⁡ x ⁢ d x = 1 2 ⁢ π ⁢ coth ⁡ ( 1 2 ⁢ π ⁢ a ) - 1 a superscript subscript 0 superscript 𝑒 𝑥 𝑎 𝑥 𝑥 𝑥 1 2 𝜋 hyperbolic-cotangent 1 2 𝜋 𝑎 1 𝑎 {\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}		 a ≠ 0 𝑎 0 {\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 ∫ 0 ∞ sinh ⁡ ( a ⁢ x ) sinh ⁡ ( π ⁢ x ) ⁢ d x = 1 2 ⁢ tan ⁡ ( 1 2 ⁢ a ) superscript subscript 0 𝑎 𝑥 𝜋 𝑥 𝑥 1 2 1 2 𝑎 {\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}		 - π < a , a < π formulae-sequence 𝜋 𝑎 𝑎 𝜋 {\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 ∫ - ∞ ∞ e a ⁢ x ( cosh ⁡ ( 1 2 ⁢ x ) ) 2 ⁢ d x = 4 ⁢ π ⁢ a sin ⁡ ( π ⁢ a ) superscript subscript superscript 𝑒 𝑎 𝑥 superscript 1 2 𝑥 2 𝑥 4 𝜋 𝑎 𝜋 𝑎 {\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}}		 - 1 < a , a < 1 formulae-sequence 1 𝑎 𝑎 1 {\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 ∫ 0 ∞ tanh ⁡ ( a ⁢ x ) - tanh ⁡ ( b ⁢ x ) x ⁢ d x = ln ⁡ ( a b ) superscript subscript 0 𝑎 𝑥 𝑏 𝑥 𝑥 𝑥 𝑎 𝑏 {\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}}		 a > 0 , b > 0 formulae-sequence 𝑎 0 𝑏 0 {\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 ∫ arcsinh ⁡ x ⁢ d x = x ⁢ arcsinh ⁡ x - ( 1 + x 2 ) 1 / 2 hyperbolic-inverse-sine 𝑥 𝑥 𝑥 hyperbolic-inverse-sine 𝑥 superscript 1 superscript 𝑥 2 1 2 {\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 ∫ arccosh ⁡ x ⁢ d x = x ⁢ arccosh ⁡ x - ( x 2 - 1 ) 1 / 2 hyperbolic-inverse-cosine 𝑥 𝑥 𝑥 hyperbolic-inverse-cosine 𝑥 superscript superscript 𝑥 2 1 1 2 {\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}		 1 < x , x < ∞ formulae-sequence 1 𝑥 𝑥 {\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 ∫ arctanh ⁡ x ⁢ d x = x ⁢ arctanh ⁡ x + 1 2 ⁢ ln ⁡ ( 1 - x 2 ) hyperbolic-inverse-tangent 𝑥 𝑥 𝑥 hyperbolic-inverse-tangent 𝑥 1 2 1 superscript 𝑥 2 {\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}}		 - 1 < x , x < 1 formulae-sequence 1 𝑥 𝑥 1 {\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 ∫ arccsch ⁡ x ⁢ d x = x ⁢ arccsch ⁡ x + arcsinh ⁡ x hyperbolic-inverse-cosecant 𝑥 𝑥 𝑥 hyperbolic-inverse-cosecant 𝑥 hyperbolic-inverse-sine 𝑥 {\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}		 0 < x , x < ∞ formulae-sequence 0 𝑥 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 ∫ arcsech ⁡ x ⁢ d x = x ⁢ arcsech ⁡ x + arcsin ⁡ x hyperbolic-inverse-secant 𝑥 𝑥 𝑥 hyperbolic-inverse-secant 𝑥 𝑥 {\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}		 0 < x , x < 1 formulae-sequence 0 𝑥 𝑥 1 {\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 ∫ arccoth ⁡ x ⁢ d x = x ⁢ arccoth ⁡ x + 1 2 ⁢ ln ⁡ ( x 2 - 1 ) hyperbolic-inverse-cotangent 𝑥 𝑥 𝑥 hyperbolic-inverse-cotangent 𝑥 1 2 superscript 𝑥 2 1 {\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}		 1 < x , x < ∞ formulae-sequence 1 𝑥 𝑥 {\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]]}

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