Optimal. Leaf size=46 \[ -\frac{1}{2} n \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} n \text{PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 x}\right ) \]
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Rubi [A] time = 0.0494245, antiderivative size = 46, 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} \[ -\frac{1}{2} n \text{PolyLog}\left (2,-e^{2 x}\right )+\frac{1}{2} n \text{PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 x}\right ) \]
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 \coth ^n(x)\right ) \, dx &=x \log \left (a \coth ^n(x)\right )+\int n x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log \left (a \coth ^n(x)\right )+n \int x \text{csch}(x) \text{sech}(x) \, dx\\ &=x \log \left (a \coth ^n(x)\right )+(2 n) \int x \text{csch}(2 x) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-n \int \log \left (1-e^{2 x}\right ) \, dx+n \int \log \left (1+e^{2 x}\right ) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac{1}{2} n \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac{1}{2} n \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=-2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac{1}{2} n \text{Li}_2\left (-e^{2 x}\right )+\frac{1}{2} n \text{Li}_2\left (e^{2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0116304, size = 55, normalized size = 1.2 \[ -\frac{1}{2} n \text{PolyLog}(2,-\tanh (x))+\frac{1}{2} n \text{PolyLog}(2,\tanh (x))-\frac{1}{2} \log (1-\tanh (x)) \log \left (a \coth ^n(x)\right )+\frac{1}{2} \log (\tanh (x)+1) \log \left (a \coth ^n(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 43, normalized size = 0.9 \begin{align*} \left ( \ln \left ( a \left ({\rm coth} \left (x\right ) \right ) ^{n} \right ) -n\ln \left ({\rm coth} \left (x\right ) \right ) \right ) x+{\frac{n{\it dilog} \left ({\rm coth} \left (x\right ) \right ) }{2}}+{\frac{n{\it dilog} \left ({\rm coth} \left (x\right )+1 \right ) }{2}}+{\frac{n\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 = 1.74351, size = 82, normalized size = 1.78 \begin{align*} -\frac{1}{2} \,{\left (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 )\right )} n + x \log \left (a \coth \left (x\right )^{n}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.92849, size = 412, normalized size = 8.96 \begin{align*} n x \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right ) + n x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - n x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - n x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + n x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + n{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - n{\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - n{\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + n{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) + x \log \left (a\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 \coth ^{n}{\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 \coth \left (x\right )^{n}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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