Optimal. Leaf size=39 \[ -\frac{1}{2} \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (\coth (x)) \]
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Rubi [A] time = 0.0458674, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 3, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.667, Rules used = {2548, 5461, 4182, 2279, 2391} \[ -\frac{1}{2} \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (\coth (x)) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 5461
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log (\coth (x)) \, dx &=x \log (\coth (x))+\int x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log (\coth (x))+2 \int x \text{csch}(2 x) \, dx\\ &=-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (\coth (x))-\int \log \left (1-e^{2 x}\right ) \, dx+\int \log \left (1+e^{2 x}\right ) \, dx\\ &=-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (\coth (x))-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=-2 x \tanh ^{-1}\left (e^{2 x}\right )+x \log (\coth (x))-\frac{1}{2} \text{Li}_2\left (-e^{2 x}\right )+\frac{\text{Li}_2\left (e^{2 x}\right )}{2}\\ \end{align*}
Mathematica [A] time = 0.0067925, size = 35, normalized size = 0.9 \[ \frac{1}{2} \text{PolyLog}(2,1-\coth (x))+\frac{1}{2} \text{PolyLog}(2,-\coth (x))+\frac{1}{2} \log (\coth (x)) \log (\coth (x)+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 24, normalized size = 0.6 \begin{align*}{\frac{{\it dilog} \left ({\rm coth} \left (x\right ) \right ) }{2}}+{\frac{{\it dilog} \left ({\rm coth} \left (x\right )+1 \right ) }{2}}+{\frac{\ln \left ({\rm coth} \left (x\right ) \right ) \ln \left ({\rm coth} \left (x\right )+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5903, size = 66, normalized size = 1.69 \begin{align*} -x \log \left (e^{\left (2 \, x\right )} + 1\right ) + x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) + x \log \left (\coth \left (x\right )\right ) - \frac{1}{2} \,{\rm Li}_2\left (-e^{\left (2 \, x\right )}\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 [C] time = 1.98114, size = 373, normalized size = 9.56 \begin{align*} x \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right ) + x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - x \log \left (-i \, \cosh \left (x\right ) - i \, \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 (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) -{\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \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 \log{\left (\coth{\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 \log \left (\coth \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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