Optimal. Leaf size=44 \[ \frac {n x^2}{2}-n x \log \left (1+e^{2 x}\right )+x \log \left (a \cosh ^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (-e^{2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 44, 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, 3799,
2221, 2317, 2438} \begin {gather*} -\frac {1}{2} n \text {PolyLog}\left (2,-e^{2 x}\right )+x \log \left (a \cosh ^n(x)\right )+\frac {n x^2}{2}-n x \log \left (e^{2 x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3799
Rubi steps
\begin {align*} \int \log \left (a \cosh ^n(x)\right ) \, dx &=x \log \left (a \cosh ^n(x)\right )-\int n x \tanh (x) \, dx\\ &=x \log \left (a \cosh ^n(x)\right )-n \int x \tanh (x) \, dx\\ &=\frac {n x^2}{2}+x \log \left (a \cosh ^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 \cosh ^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 \cosh ^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 \cosh ^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 = 0.98 \begin {gather*} \frac {1}{2} \left (-x \left (n x+2 n \log \left (1+e^{-2 x}\right )-2 \log \left (a \cosh ^n(x)\right )\right )+n \text {Li}_2\left (-e^{-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \ln \left (a \left (\cosh ^{n}\left (x \right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 36, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} n + x \log \left (a \cosh \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 = 0.40, size = 73, normalized size = 1.66 \begin {gather*} \frac {1}{2} \, n x^{2} - 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 )\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 ) + 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 \cosh ^{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 {cosh}\left (x\right )}^n\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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