Optimal. Leaf size=46 \[ -2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (-e^{2 x}\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right ) \]
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Rubi [A]
time = 0.03, 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 = {2628, 12, 5569,
4267, 2317, 2438} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2317
Rule 2438
Rule 2628
Rule 4267
Rule 5569
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 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac {1}{2} n \text {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 \begin {gather*} -\frac {1}{2} \log \left (a \coth ^n(x)\right ) \log (1-\tanh (x))+\frac {1}{2} \log \left (a \coth ^n(x)\right ) \log (1+\tanh (x))-\frac {1}{2} n \text {Li}_2(-\tanh (x))+\frac {1}{2} n \text {Li}_2(\tanh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.27, size = 43, normalized size = 0.93
method | result | size |
default | \(x \left (\ln \left (a \left (\coth ^{n}\left (x \right )\right )\right )-n \ln \left (\coth \left (x \right )\right )\right )+\frac {n \dilog \left (\coth \left (x \right )\right )}{2}+\frac {n \dilog \left (\coth \left (x \right )+1\right )}{2}+\frac {n \ln \left (\coth \left (x \right )\right ) \ln \left (\coth \left (x \right )+1\right )}{2}\) | \(43\) |
risch | \(\text {Expression too large to display}\) | \(2352\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 61, normalized size = 1.33 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.42, size = 116, normalized size = 2.52 \begin {gather*} 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 {gather*}
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 \begin {gather*} \int \log {\left (a \coth ^{n}{\left (x \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \ln \left (a\,{\mathrm {coth}\left (x\right )}^n\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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