Optimal. Leaf size=43 \[ x \log \left (a \text {sech}^n(x)\right )+\frac {1}{2} n \text {Li}_2\left (-e^{2 x}\right )-\frac {n x^2}{2}+n x \log \left (e^{2 x}+1\right ) \]
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Rubi [A] time = 0.06, 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 = {2548, 12, 3718, 2190, 2279, 2391} \[ \frac {1}{2} n \text {PolyLog}\left (2,-e^{2 x}\right )+x \log \left (a \text {sech}^n(x)\right )-\frac {n x^2}{2}+n x \log \left (e^{2 x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 2548
Rule 3718
Rubi steps
\begin {align*} \int \log \left (a \text {sech}^n(x)\right ) \, dx &=x \log \left (a \text {sech}^n(x)\right )+\int n x \tanh (x) \, dx\\ &=x \log \left (a \text {sech}^n(x)\right )+n \int x \tanh (x) \, dx\\ &=-\frac {n x^2}{2}+x \log \left (a \text {sech}^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 {sech}^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 {sech}^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 \text {sech}^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 \[ x \log \left (a \text {sech}^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (-e^{-2 x}\right )+\frac {n x^2}{2}+n x \log \left (e^{-2 x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.46, size = 92, normalized size = 2.14 \[ -\frac {1}{2} \, n x^{2} + n x \log \left (\frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}\right ) + n x \log \left (i \, \cosh \relax (x) + i \, \sinh \relax (x) + 1\right ) + n x \log \left (-i \, \cosh \relax (x) - i \, \sinh \relax (x) + 1\right ) + n {\rm Li}_2\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right ) + n {\rm Li}_2\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right ) + x \log \relax (a) \]
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 \[ \int \log \left (a \operatorname {sech}\relax (x)^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.86, size = 0, normalized size = 0.00 \[ \int \ln \left (a \mathrm {sech}\relax (x )^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 36, normalized size = 0.84 \[ -\frac {1}{2} \, {\left (x^{2} - 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} n + x \log \left (a \operatorname {sech}\relax (x)^{n}\right ) \]
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 \[ \int \ln \left (a\,{\left (\frac {1}{\mathrm {cosh}\relax (x)}\right )}^n\right ) \,d x \]
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 \[ \int \log {\left (a \operatorname {sech}^{n}{\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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