Optimal. Leaf size=39 \[ -\frac{1}{2} \text{PolyLog}\left (2,-e^{2 x}\right )+x \log (a \cosh (x))+\frac{x^2}{2}-x \log \left (e^{2 x}+1\right ) \]
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Rubi [A] time = 0.0556963, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2548, 3718, 2190, 2279, 2391} \[ -\frac{1}{2} \text{PolyLog}\left (2,-e^{2 x}\right )+x \log (a \cosh (x))+\frac{x^2}{2}-x \log \left (e^{2 x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log (a \cosh (x)) \, dx &=x \log (a \cosh (x))-\int x \tanh (x) \, dx\\ &=\frac{x^2}{2}+x \log (a \cosh (x))-2 \int \frac{e^{2 x} x}{1+e^{2 x}} \, dx\\ &=\frac{x^2}{2}-x \log \left (1+e^{2 x}\right )+x \log (a \cosh (x))+\int \log \left (1+e^{2 x}\right ) \, dx\\ &=\frac{x^2}{2}-x \log \left (1+e^{2 x}\right )+x \log (a \cosh (x))+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac{x^2}{2}-x \log \left (1+e^{2 x}\right )+x \log (a \cosh (x))-\frac{1}{2} \text{Li}_2\left (-e^{2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0183128, size = 36, normalized size = 0.92 \[ \frac{1}{2} \left (\text{PolyLog}\left (2,-e^{-2 x}\right )-x \left (-2 \log (a \cosh (x))+x+2 \log \left (e^{-2 x}+1\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.109, size = 321, normalized size = 8.2 \begin{align*} -x\ln \left ({{\rm e}^{x}} \right ) +{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{-x}} \left ( 1+{{\rm e}^{2\,x}} \right ) \right ) \left ({\it csgn} \left ( ia \left ( 1+{{\rm e}^{2\,x}} \right ){{\rm e}^{-x}} \right ) \right ) ^{2}x+{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ia \left ( 1+{{\rm e}^{2\,x}} \right ){{\rm e}^{-x}} \right ) \right ) ^{2}{\it csgn} \left ( ia \right ) x-{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{-x}} \left ( 1+{{\rm e}^{2\,x}} \right ) \right ){\it csgn} \left ( ia \left ( 1+{{\rm e}^{2\,x}} \right ){{\rm e}^{-x}} \right ){\it csgn} \left ( ia \right ) x+{\frac{i}{2}}\pi \,{\it csgn} \left ( i \left ( 1+{{\rm e}^{2\,x}} \right ) \right ) \left ({\it csgn} \left ( i{{\rm e}^{-x}} \left ( 1+{{\rm e}^{2\,x}} \right ) \right ) \right ) ^{2}x-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( i{{\rm e}^{-x}} \left ( 1+{{\rm e}^{2\,x}} \right ) \right ) \right ) ^{3}x+{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{-x}} \right ) \left ({\it csgn} \left ( i{{\rm e}^{-x}} \left ( 1+{{\rm e}^{2\,x}} \right ) \right ) \right ) ^{2}x+\ln \left ( a \right ) x-\ln \left ( 2 \right ) x+{\frac{{x}^{2}}{2}}-{\frac{i}{2}}\pi \, \left ({\it csgn} \left ( ia \left ( 1+{{\rm e}^{2\,x}} \right ){{\rm e}^{-x}} \right ) \right ) ^{3}x-{\frac{i}{2}}\pi \,{\it csgn} \left ( i{{\rm e}^{-x}} \right ){\it csgn} \left ( i \left ( 1+{{\rm e}^{2\,x}} \right ) \right ){\it csgn} \left ( i{{\rm e}^{-x}} \left ( 1+{{\rm e}^{2\,x}} \right ) \right ) x+\ln \left ({{\rm e}^{x}} \right ) \ln \left ( 1+{{\rm e}^{2\,x}} \right ) -\ln \left ({{\rm e}^{x}} \right ) \ln \left ( 1+i{{\rm e}^{x}} \right ) -\ln \left ({{\rm e}^{x}} \right ) \ln \left ( 1-i{{\rm e}^{x}} \right ) -{\it dilog} \left ( 1+i{{\rm e}^{x}} \right ) -{\it dilog} \left ( 1-i{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70867, size = 43, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} + x \log \left (a \cosh \left (x\right )\right ) - x \log \left (e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{2} \,{\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 = 2.00195, size = 219, normalized size = 5.62 \begin{align*} \frac{1}{2} \, x^{2} + x \log \left (a \cosh \left (x\right )\right ) - x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) -{\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) -{\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 \cosh{\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 )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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