Optimal. Leaf size=27 \[ -\text {Li}_2\left (-e^x\right )+\text {Li}_2\left (e^x\right )-2 x \tanh ^{-1}\left (e^x\right )+x \coth ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 2279
Rule 2391
Rule 4182
Rule 6272
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*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.74 \[ \text {Li}_2\left (-e^{-x}\right )-\text {Li}_2\left (e^{-x}\right )+x \left (\log \left (1-e^{-x}\right )-\log \left (e^{-x}+1\right )\right )+x \coth ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.39, size = 57, normalized size = 2.11 \[ \frac {1}{2} \, x \log \left (\frac {\cosh \relax (x) + 1}{\cosh \relax (x) - 1}\right ) - x \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + x \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcoth}\left (\cosh \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 21, normalized size = 0.78 \[ x \,\mathrm {arccoth}\left (\cosh \relax (x )\right )+2 \dilog \left ({\mathrm e}^{-x}\right )-\frac {\dilog \left ({\mathrm e}^{-2 x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 33, normalized size = 1.22 \[ x \operatorname {arcoth}\left (\cosh \relax (x)\right ) - x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) - {\rm Li}_2\left (-e^{x}\right ) + {\rm Li}_2\left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \mathrm {acoth}\left (\mathrm {cosh}\relax (x)\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acoth}{\left (\cosh {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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