Optimal. Leaf size=27 \[ -\text{PolyLog}\left (2,-e^x\right )+\text{PolyLog}\left (2,e^x\right )-2 x \tanh ^{-1}\left (e^x\right )+x \coth ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0342236, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 3, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.333, Rules used = {6272, 4182, 2279, 2391} \[ -\text{PolyLog}\left (2,-e^x\right )+\text{PolyLog}\left (2,e^x\right )-2 x \tanh ^{-1}\left (e^x\right )+x \coth ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 6272
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \coth ^{-1}(\cosh (x)) \, dx &=x \coth ^{-1}(\cosh (x))+\int x \text{csch}(x) \, dx\\ &=x \coth ^{-1}(\cosh (x))-2 x \tanh ^{-1}\left (e^x\right )-\int \log \left (1-e^x\right ) \, dx+\int \log \left (1+e^x\right ) \, dx\\ &=x \coth ^{-1}(\cosh (x))-2 x \tanh ^{-1}\left (e^x\right )-\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^x\right )+\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^x\right )\\ &=x \coth ^{-1}(\cosh (x))-2 x \tanh ^{-1}\left (e^x\right )-\text{Li}_2\left (-e^x\right )+\text{Li}_2\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0196667, size = 47, normalized size = 1.74 \[ \text{PolyLog}\left (2,-e^{-x}\right )-\text{PolyLog}\left (2,e^{-x}\right )+x \left (\log \left (1-e^{-x}\right )-\log \left (e^{-x}+1\right )\right )+x \coth ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 21, normalized size = 0.8 \begin{align*} x{\rm arccoth} \left (\cosh \left ( x \right ) \right )+2\,{\it dilog} \left ({{\rm e}^{-x}} \right ) -{\frac{{\it dilog} \left ({{\rm e}^{-2\,x}} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15314, size = 45, normalized size = 1.67 \begin{align*} x \operatorname{arcoth}\left (\cosh \left (x\right )\right ) - x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) -{\rm Li}_2\left (-e^{x}\right ) +{\rm Li}_2\left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66712, size = 213, normalized size = 7.89 \begin{align*} \frac{1}{2} \, x \log \left (\frac{\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) +{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) -{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acoth}{\left (\cosh{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcoth}\left (\cosh \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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