Optimal. Leaf size=51 \[ -x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Li}_3\left (-e^x\right )-\text {Li}_3\left (e^x\right )-x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \coth ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.07, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6274, 4182, 2531, 2282, 6589} \[ -x \text {PolyLog}\left (2,-e^x\right )+x \text {PolyLog}\left (2,e^x\right )+\text {PolyLog}\left (3,-e^x\right )-\text {PolyLog}\left (3,e^x\right )-x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \coth ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4182
Rule 6274
Rule 6589
Rubi steps
\begin {align*} \int x \coth ^{-1}(\cosh (x)) \, dx &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))+\frac {1}{2} \int x^2 \text {csch}(x) \, dx\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-\int x \log \left (1-e^x\right ) \, dx+\int x \log \left (1+e^x\right ) \, dx\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\int \text {Li}_2\left (-e^x\right ) \, dx-\int \text {Li}_2\left (e^x\right ) \, dx\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^x\right )\\ &=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \tanh ^{-1}\left (e^x\right )-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Li}_3\left (-e^x\right )-\text {Li}_3\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 1.59 \[ \frac {1}{2} \left (2 x \text {Li}_2\left (-e^{-x}\right )-2 x \text {Li}_2\left (e^{-x}\right )+2 \text {Li}_3\left (-e^{-x}\right )-2 \text {Li}_3\left (e^{-x}\right )+x^2 \log \left (1-e^{-x}\right )-x^2 \log \left (e^{-x}+1\right )+x^2 \coth ^{-1}(\cosh (x))\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 2.01, size = 87, normalized size = 1.71 \[ \frac {1}{4} \, x^{2} \log \left (\frac {\cosh \relax (x) + 1}{\cosh \relax (x) - 1}\right ) - \frac {1}{2} \, x^{2} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + x {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - x {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) - {\rm polylog}\left (3, \cosh \relax (x) + \sinh \relax (x)\right ) + {\rm polylog}\left (3, -\cosh \relax (x) - \sinh \relax (x)\right ) \]
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 \[ \int x \operatorname {arcoth}\left (\cosh \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.52, size = 449, normalized size = 8.80 \[ -\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{3} x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right ) x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right )^{3} x^{2}}{8}+\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) x^{2}}{8}+\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{4}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{4}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right ) x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{8}+\polylog \left (3, -{\mathrm e}^{x}\right )-\polylog \left (3, {\mathrm e}^{x}\right )+\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x^{2}}{8}-\frac {x^{2} \ln \left ({\mathrm e}^{x}-1\right )}{2}+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{2}+x \polylog \left (2, {\mathrm e}^{x}\right )-x \polylog \left (2, -{\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 56, normalized size = 1.10 \[ \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (\cosh \relax (x)\right ) - \frac {1}{2} \, x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-e^{x} + 1\right ) - x {\rm Li}_2\left (-e^{x}\right ) + x {\rm Li}_2\left (e^{x}\right ) + {\rm Li}_{3}(-e^{x}) - {\rm Li}_{3}(e^{x}) \]
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 \[ \int x\,\mathrm {acoth}\left (\mathrm {cosh}\relax (x)\right ) \,d x \]
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 \[ \int x \operatorname {acoth}{\left (\cosh {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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