Optimal. Leaf size=38 \[ \frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )+x \log (a \text{csch}(x))-\frac{x^2}{2}+x \log \left (1-e^{2 x}\right ) \]
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Rubi [A] time = 0.0539876, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2548, 3716, 2190, 2279, 2391} \[ \frac{1}{2} \text{PolyLog}\left (2,e^{2 x}\right )+x \log (a \text{csch}(x))-\frac{x^2}{2}+x \log \left (1-e^{2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log (a \text{csch}(x)) \, dx &=x \log (a \text{csch}(x))+\int x \coth (x) \, dx\\ &=-\frac{x^2}{2}+x \log (a \text{csch}(x))-2 \int \frac{e^{2 x} x}{1-e^{2 x}} \, dx\\ &=-\frac{x^2}{2}+x \log \left (1-e^{2 x}\right )+x \log (a \text{csch}(x))-\int \log \left (1-e^{2 x}\right ) \, dx\\ &=-\frac{x^2}{2}+x \log \left (1-e^{2 x}\right )+x \log (a \text{csch}(x))-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )\\ &=-\frac{x^2}{2}+x \log \left (1-e^{2 x}\right )+x \log (a \text{csch}(x))+\frac{\text{Li}_2\left (e^{2 x}\right )}{2}\\ \end{align*}
Mathematica [A] time = 0.0155529, size = 37, normalized size = 0.97 \[ \frac{1}{2} \left (x \left (2 \log (a \text{csch}(x))+x+2 \log \left (1-e^{-2 x}\right )\right )-\text{PolyLog}\left (2,e^{-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.102, size = 293, normalized size = 7.7 \begin{align*} x\ln \left ({{\rm e}^{x}} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ({\frac{i{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ){\it csgn} \left ({\frac{ia{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ){\it csgn} \left ( ia \right ) x+{\frac{i}{2}}\pi \,{\it csgn} \left ({\frac{i{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) \left ({\it csgn} \left ({\frac{ia{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) \right ) ^{2}x-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ({\frac{i{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) \right ) ^{3}x-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ({\frac{ia{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) \right ) ^{3}x+{\frac{i}{2}}\pi \,{\it csgn} \left ({\frac{i}{{{\rm e}^{2\,x}}-1}} \right ) \left ({\it csgn} \left ({\frac{i{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) \right ) ^{2}x+{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{x}} \right ) \left ({\it csgn} \left ({\frac{i{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) \right ) ^{2}x+\ln \left ( 2 \right ) x+\ln \left ( a \right ) x-{\frac{{x}^{2}}{2}}+{\frac{i}{2}}\pi \, \left ({\it csgn} \left ({\frac{ia{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) \right ) ^{2}{\it csgn} \left ( ia \right ) x-{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{x}} \right ){\it csgn} \left ({\frac{i}{{{\rm e}^{2\,x}}-1}} \right ){\it csgn} \left ({\frac{i{{\rm e}^{x}}}{{{\rm e}^{2\,x}}-1}} \right ) x-\ln \left ({{\rm e}^{x}} \right ) \ln \left ({{\rm e}^{2\,x}}-1 \right ) -{\it dilog} \left ({{\rm e}^{x}} \right ) +{\it dilog} \left ({{\rm e}^{x}}+1 \right ) +\ln \left ({{\rm e}^{x}} \right ) \ln \left ({{\rm e}^{x}}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21529, size = 50, normalized size = 1.32 \begin{align*} -\frac{1}{2} \, x^{2} + x \log \left (a \operatorname{csch}\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 = 1.88009, size = 285, normalized size = 7.5 \begin{align*} -\frac{1}{2} \, x^{2} + x \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 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 \log{\left (a \operatorname{csch}{\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 (a \operatorname{csch}\left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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