Optimal. Leaf size=44 \[ -\frac{1}{2} n \text{PolyLog}\left (2,-e^{2 x}\right )+x \log \left (a \cosh ^n(x)\right )+\frac{n x^2}{2}-n x \log \left (e^{2 x}+1\right ) \]
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Rubi [A] time = 0.0566674, antiderivative size = 44, normalized size of antiderivative = 1., 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 = {2548, 12, 3718, 2190, 2279, 2391} \[ -\frac{1}{2} n \text{PolyLog}\left (2,-e^{2 x}\right )+x \log \left (a \cosh ^n(x)\right )+\frac{n x^2}{2}-n x \log \left (e^{2 x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 12
Rule 3718
Rule 2190
Rule 2279
Rule 2391
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 \operatorname{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*}
Mathematica [A] time = 0.028383, size = 43, normalized size = 0.98 \[ \frac{1}{2} \left (n \text{PolyLog}\left (2,-e^{-2 x}\right )-x \left (-2 \log \left (a \cosh ^n(x)\right )+n x+2 n \log \left (e^{-2 x}+1\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( a \left ( \cosh \left ( x \right ) \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67901, size = 49, normalized size = 1.11 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.08226, size = 247, normalized size = 5.61 \begin{align*} \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{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 \cosh ^{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 \cosh \left (x\right )^{n}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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