View source for 4.40

{{DISPLAYTITLE:Elementary Functions - 4.40 Integrals}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/4.40.E1 4.40.E1] || [[Item:Q1973|<math>\int\sinh@@{x}\diff{x} = \cosh@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\sinh@@{x}\diff{x} = \cosh@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(sinh(x), x) = cosh(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Sinh[x], x, GenerateConditions->None] == Cosh[x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E2 4.40.E2] || [[Item:Q1974|<math>\int\cosh@@{x}\diff{x} = \sinh@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\cosh@@{x}\diff{x} = \sinh@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(cosh(x), x) = sinh(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Cosh[x], x, GenerateConditions->None] == Sinh[x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E3 4.40.E3] || [[Item:Q1975|<math>\int\tanh@@{x}\diff{x} = \ln@{\cosh@@{x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\tanh@@{x}\diff{x} = \ln@{\cosh@@{x}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(tanh(x), x) = ln(cosh(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Tanh[x], x, GenerateConditions->None] == Log[Cosh[x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E4 4.40.E4] || [[Item:Q1976|<math>\int\csch@@{x}\diff{x} = \ln@{\tanh@{\tfrac{1}{2}x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\csch@@{x}\diff{x} = \ln@{\tanh@{\tfrac{1}{2}x}}</syntaxhighlight> || <math>0 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(csch(x), x) = ln(tanh((1)/(2)*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Csch[x], x, GenerateConditions->None] == Log[Tanh[Divide[1,2]*x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E5 4.40.E5] || [[Item:Q1977|<math>\int\sech@@{x}\diff{x} = \Gudermannian@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\sech@@{x}\diff{x} = \Gudermannian@{x}</syntaxhighlight> || <math>-\infty < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(sech(x), x) = arctan(sinh(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Sech[x], x, GenerateConditions->None] == Gudermannian[x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E6 4.40.E6] || [[Item:Q1978|<math>\int\coth@@{x}\diff{x} = \ln@{\sinh@@{x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\coth@@{x}\diff{x} = \ln@{\sinh@@{x}}</syntaxhighlight> || <math>0 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(coth(x), x) = ln(sinh(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Coth[x], x, GenerateConditions->None] == Log[Sinh[x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E7 4.40.E7] || [[Item:Q1979|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math>a \neq 0</math> || <syntaxhighlight lang=mathematica>int(exp(- x)*(sin(a*x))/(sinh(x)), x = 0..infinity) = (1)/(2)*Pi*coth((1)/(2)*Pi*a)-(1)/(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 6] || Successful [Tested: 6]
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| [https://dlmf.nist.gov/4.40.E8 4.40.E8] || [[Item:Q1980|<math>\int_{0}^{\infty}\frac{\sinh@{ax}}{\sinh@{\pi x}}\diff{x} = \tfrac{1}{2}\tan@{\tfrac{1}{2}a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{\sinh@{ax}}{\sinh@{\pi x}}\diff{x} = \tfrac{1}{2}\tan@{\tfrac{1}{2}a}</syntaxhighlight> || <math>-\pi < a, a < \pi</math> || <syntaxhighlight lang=mathematica>int((sinh(a*x))/(sinh(Pi*x)), x = 0..infinity) = (1)/(2)*tan((1)/(2)*a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[Sinh[a*x],Sinh[Pi*x]], {x, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Tan[Divide[1,2]*a]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 6] || Skipped - Because timed out
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| [https://dlmf.nist.gov/4.40.E9 4.40.E9] || [[Item:Q1981|<math>\int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh@{\tfrac{1}{2}x}\right)^{2}}\diff{x} = \frac{4\pi a}{\sin@{\pi a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh@{\tfrac{1}{2}x}\right)^{2}}\diff{x} = \frac{4\pi a}{\sin@{\pi a}}</syntaxhighlight> || <math>-1 < a, a < 1</math> || <syntaxhighlight lang=mathematica>int((exp(a*x))/((cosh((1)/(2)*x))^(2)), x = - infinity..infinity) = (4*Pi*a)/(sin(Pi*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[Exp[a*x],(Cosh[Divide[1,2]*x])^(2)], {x, - Infinity, Infinity}, GenerateConditions->None] == Divide[4*Pi*a,Sin[Pi*a]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.40.E10 4.40.E10] || [[Item:Q1982|<math>\int_{0}^{\infty}\frac{\tanh@{ax}-\tanh@{bx}}{x}\diff{x} = \ln@{\frac{a}{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{\tanh@{ax}-\tanh@{bx}}{x}\diff{x} = \ln@{\frac{a}{b}}</syntaxhighlight> || <math>a > 0, b > 0</math> || <syntaxhighlight lang=mathematica>int((tanh(a*x)- tanh(b*x))/(x), x = 0..infinity) = ln((a)/(b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[Tanh[a*x]- Tanh[b*x],x], {x, 0, Infinity}, GenerateConditions->None] == Log[Divide[a,b]]</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
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| [https://dlmf.nist.gov/4.40.E11 4.40.E11] || [[Item:Q1983|<math>\int\asinh@@{x}\diff{x} = x\asinh@@{x}-(1+x^{2})^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\asinh@@{x}\diff{x} = x\asinh@@{x}-(1+x^{2})^{1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(arcsinh(x), x) = x*arcsinh(x)-(1 + (x)^(2))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcSinh[x], x, GenerateConditions->None] == x*ArcSinh[x]-(1 + (x)^(2))^(1/2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E12 4.40.E12] || [[Item:Q1984|<math>\int\acosh@@{x}\diff{x} = x\acosh@@{x}-(x^{2}-1)^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\acosh@@{x}\diff{x} = x\acosh@@{x}-(x^{2}-1)^{1/2}</syntaxhighlight> || <math>1 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(arccosh(x), x) = x*arccosh(x)-((x)^(2)- 1)^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcCosh[x], x, GenerateConditions->None] == x*ArcCosh[x]-((x)^(2)- 1)^(1/2)</syntaxhighlight> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/4.40.E13 4.40.E13] || [[Item:Q1985|<math>\int\atanh@@{x}\diff{x} = x\atanh@@{x}+\tfrac{1}{2}\ln@{1-x^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\atanh@@{x}\diff{x} = x\atanh@@{x}+\tfrac{1}{2}\ln@{1-x^{2}}</syntaxhighlight> || <math>-1 < x, x < 1</math> || <syntaxhighlight lang=mathematica>int(arctanh(x), x) = x*arctanh(x)+(1)/(2)*ln(1 - (x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcTanh[x], x, GenerateConditions->None] == x*ArcTanh[x]+Divide[1,2]*Log[1 - (x)^(2)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.40.E14 4.40.E14] || [[Item:Q1986|<math>\int\acsch@@{x}\diff{x} = x\acsch@@{x}+\asinh@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\acsch@@{x}\diff{x} = x\acsch@@{x}+\asinh@@{x}</syntaxhighlight> || <math>0 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(arccsch(x), x) = x*arccsch(x)+ arcsinh(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcCsch[x], x, GenerateConditions->None] == x*ArcCsch[x]+ ArcSinh[x]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/4.40.E15 4.40.E15] || [[Item:Q1987|<math>\int\asech@@{x}\diff{x} = x\asech@@{x}+\asin@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\asech@@{x}\diff{x} = x\asech@@{x}+\asin@@{x}</syntaxhighlight> || <math>0 < x, x < 1</math> || <syntaxhighlight lang=mathematica>int(arcsech(x), x) = x*arcsech(x)+ arcsin(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcSech[x], x, GenerateConditions->None] == x*ArcSech[x]+ ArcSin[x]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.570796327
Test Values: {x = .5}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
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| [https://dlmf.nist.gov/4.40.E16 4.40.E16] || [[Item:Q1988|<math>\int\acoth@@{x}\diff{x} = x\acoth@@{x}+\tfrac{1}{2}\ln@{x^{2}-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int\acoth@@{x}\diff{x} = x\acoth@@{x}+\tfrac{1}{2}\ln@{x^{2}-1}</syntaxhighlight> || <math>1 < x, x < \infty</math> || <syntaxhighlight lang=mathematica>int(arccoth(x), x) = x*arccoth(x)+(1)/(2)*ln((x)^(2)- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ArcCoth[x], x, GenerateConditions->None] == x*ArcCoth[x]+Divide[1,2]*Log[(x)^(2)- 1]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0, -1.5707963267948966]
Test Values: {Rule[x, Rational[1, 2]]}</syntaxhighlight><br></div></div>
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