Optimal. Leaf size=35 \[ \text{PolyLog}\left (2,-e^{2 x}\right )+x \log \left (a \text{sech}^2(x)\right )-x^2+2 x \log \left (e^{2 x}+1\right ) \]
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Rubi [A] time = 0.0534823, antiderivative size = 35, 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} \[ \text{PolyLog}\left (2,-e^{2 x}\right )+x \log \left (a \text{sech}^2(x)\right )-x^2+2 x \log \left (e^{2 x}+1\right ) \]
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 \text{sech}^2(x)\right ) \, dx &=x \log \left (a \text{sech}^2(x)\right )-\int -2 x \tanh (x) \, dx\\ &=x \log \left (a \text{sech}^2(x)\right )+2 \int x \tanh (x) \, dx\\ &=-x^2+x \log \left (a \text{sech}^2(x)\right )+4 \int \frac{e^{2 x} x}{1+e^{2 x}} \, dx\\ &=-x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text{sech}^2(x)\right )-2 \int \log \left (1+e^{2 x}\right ) \, dx\\ &=-x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text{sech}^2(x)\right )-\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=-x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text{sech}^2(x)\right )+\text{Li}_2\left (-e^{2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.017846, size = 33, normalized size = 0.94 \[ x \left (\log \left (a \text{sech}^2(x)\right )+x+2 \log \left (e^{-2 x}+1\right )\right )-\text{PolyLog}\left (2,-e^{-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.142, size = 480, normalized size = 13.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67255, size = 43, normalized size = 1.23 \begin{align*} -x^{2} + x \log \left (a \operatorname{sech}\left (x\right )^{2}\right ) + 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.04056, size = 363, normalized size = 10.37 \begin{align*} -x^{2} + x \log \left (\frac{4 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + 3 \, \cosh \left (x\right )}\right ) + 2 \, x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + 2 \, x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + 2 \,{\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + 2 \,{\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\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 \operatorname{sech}^{2}{\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 \operatorname{sech}\left (x\right )^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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