Optimal. Leaf size=37 \[ \text{PolyLog}\left (2,-e^{2 x}\right )-\text{PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \tanh ^2(x)\right )+4 x \tanh ^{-1}\left (e^{2 x}\right ) \]
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Rubi [A] time = 0.0470444, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {2548, 12, 5461, 4182, 2279, 2391} \[ \text{PolyLog}\left (2,-e^{2 x}\right )-\text{PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \tanh ^2(x)\right )+4 x \tanh ^{-1}\left (e^{2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 12
Rule 5461
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log \left (a \tanh ^2(x)\right ) \, dx &=x \log \left (a \tanh ^2(x)\right )-\int 2 x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log \left (a \tanh ^2(x)\right )-2 \int x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log \left (a \tanh ^2(x)\right )-4 \int x \text{csch}(2 x) \, dx\\ &=4 x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \tanh ^2(x)\right )+2 \int \log \left (1-e^{2 x}\right ) \, dx-2 \int \log \left (1+e^{2 x}\right ) \, dx\\ &=4 x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \tanh ^2(x)\right )+\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )-\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=4 x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \tanh ^2(x)\right )+\text{Li}_2\left (-e^{2 x}\right )-\text{Li}_2\left (e^{2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0095761, size = 47, normalized size = 1.27 \[ \text{PolyLog}(2,-\tanh (x))-\text{PolyLog}(2,\tanh (x))-\frac{1}{2} \log (1-\tanh (x)) \log \left (a \tanh ^2(x)\right )+\frac{1}{2} \log (\tanh (x)+1) \log \left (a \tanh ^2(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 47, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( \tanh \left ( x \right ) -1 \right ) \ln \left ( a \left ( \tanh \left ( x \right ) \right ) ^{2} \right ) }{2}}+{\it dilog} \left ( \tanh \left ( x \right ) \right ) +\ln \left ( \tanh \left ( x \right ) -1 \right ) \ln \left ( \tanh \left ( x \right ) \right ) +{\frac{\ln \left ( \tanh \left ( x \right ) +1 \right ) \ln \left ( a \left ( \tanh \left ( x \right ) \right ) ^{2} \right ) }{2}}+{\it dilog} \left ( \tanh \left ( x \right ) +1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54846, size = 77, normalized size = 2.08 \begin{align*} x \log \left (a \tanh \left (x\right )^{2}\right ) + 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - 2 \,{\rm Li}_2\left (-e^{x}\right ) - 2 \,{\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 = 1.969, size = 454, normalized size = 12.27 \begin{align*} x \log \left (\frac{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} - a}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 1}\right ) - 2 \, x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \, x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + 2 \, x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) - 2 \, x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) - 2 \,{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \,{\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + 2 \,{\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) - 2 \,{\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 \tanh ^{2}{\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 \tanh \left (x\right )^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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