Optimal. Leaf size=39 \[ x \log (a \sinh (x))-\frac {\text {Li}_2\left (e^{2 x}\right )}{2}+\frac {x^2}{2}-x \log \left (1-e^{2 x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2548, 3716, 2190, 2279, 2391} \[ -\frac {1}{2} \text {PolyLog}\left (2,e^{2 x}\right )+x \log (a \sinh (x))+\frac {x^2}{2}-x \log \left (1-e^{2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 2548
Rule 3716
Rubi steps
\begin {align*} \int \log (a \sinh (x)) \, dx &=x \log (a \sinh (x))-\int x \coth (x) \, dx\\ &=\frac {x^2}{2}+x \log (a \sinh (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 \sinh (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 \sinh (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 \sinh (x))-\frac {\text {Li}_2\left (e^{2 x}\right )}{2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.92 \[ \frac {1}{2} \left (\text {Li}_2\left (e^{-2 x}\right )-x \left (-2 \log (a \sinh (x))+x+2 \log \left (1-e^{-2 x}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 57, normalized size = 1.46 \[ \frac {1}{2} \, x^{2} + x \log \left (a \sinh \relax (x)\right ) - x \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - x \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) - {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) \]
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 \[ \int \log \left (a \sinh \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.99, size = 295, normalized size = 7.56 \[ -\frac {i \pi x \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) a \,{\mathrm e}^{-x}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) a \,{\mathrm e}^{-x}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x}\right )^{3}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) a \,{\mathrm e}^{-x}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}-1\right ) a \,{\mathrm e}^{-x}\right )^{3}}{2}+\frac {x^{2}}{2}+x \ln \relax (a )-x \ln \left ({\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}\right )-\ln \relax (2) x -\dilog \left ({\mathrm e}^{x}+1\right )+\dilog \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 43, normalized size = 1.10 \[ \frac {1}{2} \, x^{2} + x \log \left (a \sinh \relax (x)\right ) - x \log \left (e^{x} + 1\right ) - x \log \left (-e^{x} + 1\right ) - {\rm Li}_2\left (-e^{x}\right ) - {\rm Li}_2\left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \ln \left (a\,\mathrm {sinh}\relax (x)\right ) \,d x \]
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 \[ \int \log {\left (a \sinh {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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