Optimal. Leaf size=46 \[ x \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (-e^{2 x}\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right )-2 n x \tanh ^{-1}\left (e^{2 x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 12
Rule 2279
Rule 2391
Rule 2548
Rule 4182
Rule 5461
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*}
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Mathematica [A] time = 0.01, size = 55, normalized size = 1.20 \[ -\frac {1}{2} \log (1-\tanh (x)) \log \left (a \coth ^n(x)\right )+\frac {1}{2} \log (\tanh (x)+1) \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \text {Li}_2(-\tanh (x))+\frac {1}{2} n \text {Li}_2(\tanh (x)) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.49, size = 116, normalized size = 2.52 \[ n x \log \left (\frac {\cosh \relax (x)}{\sinh \relax (x)}\right ) + n x \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - n x \log \left (i \, \cosh \relax (x) + i \, \sinh \relax (x) + 1\right ) - n x \log \left (-i \, \cosh \relax (x) - i \, \sinh \relax (x) + 1\right ) + n x \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + n {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - n {\rm Li}_2\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right ) - n {\rm Li}_2\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right ) + n {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) + x \log \relax (a) \]
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 \[ \int \log \left (a \coth \relax (x)^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 43, normalized size = 0.93 \[ \frac {n \ln \left (\coth \relax (x )+1\right ) \ln \left (\coth \relax (x )\right )}{2}+\frac {n \dilog \left (\coth \relax (x )+1\right )}{2}+\frac {n \dilog \left (\coth \relax (x )\right )}{2}+\left (-n \ln \left (\coth \relax (x )\right )+\ln \left (a \left (\coth ^{n}\relax (x )\right )\right )\right ) x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 61, normalized size = 1.33 \[ -\frac {1}{2} \, {\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - 2 \, {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_2\left (e^{x}\right )\right )} n + x \log \left (a \coth \relax (x)^{n}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \ln \left (a\,{\mathrm {coth}\relax (x)}^n\right ) \,d x \]
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 \[ \int \log {\left (a \coth ^{n}{\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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