Optimal. Leaf size=43 \[ -\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 3797,
2221, 2317, 2438} \begin {gather*} \frac {1}{2} n \text {PolyLog}\left (2,e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3797
Rubi steps
\begin {align*} \int \log \left (a \text {csch}^n(x)\right ) \, dx &=x \log \left (a \text {csch}^n(x)\right )+\int n x \coth (x) \, dx\\ &=x \log \left (a \text {csch}^n(x)\right )+n \int x \coth (x) \, dx\\ &=-\frac {n x^2}{2}+x \log \left (a \text {csch}^n(x)\right )-(2 n) \int \frac {e^{2 x} x}{1-e^{2 x}} \, dx\\ &=-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-n \int \log \left (1-e^{2 x}\right ) \, dx\\ &=-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-\frac {1}{2} n \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right )\\ &=-\frac {n x^2}{2}+n x \log \left (1-e^{2 x}\right )+x \log \left (a \text {csch}^n(x)\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.00 \begin {gather*} \frac {n x^2}{2}+n x \log \left (1-e^{-2 x}\right )+x \log \left (a \text {csch}^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (e^{-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \ln \left (a \mathrm {csch}\left (x \right )^{n}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 47, normalized size = 1.09 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) - 2 \, {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_2\left (e^{x}\right )\right )} n + x \log \left (a \operatorname {csch}\left (x\right )^{n}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs.
\(2 (36) = 72\).
time = 0.41, size = 84, normalized size = 1.95 \begin {gather*} -\frac {1}{2} \, n x^{2} + n x \log \left (\frac {2 \, {\left (\cosh \left (x\right ) + \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 ) + n x \log \left (\cosh \left (x\right ) + \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 (-\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 \operatorname {csch}^{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\,{\left (\frac {1}{\mathrm {sinh}\left (x\right )}\right )}^n\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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