Optimal. Leaf size=51 \[ \frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )+\frac {\text {Li}_3\left (-e^{-x}\right )}{2}-\frac {\text {Li}_3\left (e^{-x}\right )}{2} \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6214, 2531, 2282, 6589} \[ \frac {1}{2} x \text {PolyLog}\left (2,-e^{-x}\right )-\frac {1}{2} x \text {PolyLog}\left (2,e^{-x}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{-x}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{-x}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 6214
Rule 6589
Rubi steps
\begin {align*} \int x \coth ^{-1}\left (e^x\right ) \, dx &=-\left (\frac {1}{2} \int x \log \left (1-e^{-x}\right ) \, dx\right )+\frac {1}{2} \int x \log \left (1+e^{-x}\right ) \, dx\\ &=\frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )-\frac {1}{2} \int \text {Li}_2\left (-e^{-x}\right ) \, dx+\frac {1}{2} \int \text {Li}_2\left (e^{-x}\right ) \, dx\\ &=\frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{-x}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{-x}\right )\\ &=\frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )+\frac {\text {Li}_3\left (-e^{-x}\right )}{2}-\frac {\text {Li}_3\left (e^{-x}\right )}{2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 1.39 \[ \frac {1}{4} \left (-2 x \text {Li}_2\left (-e^x\right )+2 x \text {Li}_2\left (e^x\right )+2 \text {Li}_3\left (-e^x\right )-2 \text {Li}_3\left (e^x\right )+x^2 \log \left (1-e^x\right )-x^2 \log \left (e^x+1\right )+2 x^2 \coth ^{-1}\left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.58, size = 94, normalized size = 1.84 \[ \frac {1}{4} \, x^{2} \log \left (\frac {\cosh \relax (x) + \sinh \relax (x) + 1}{\cosh \relax (x) + \sinh \relax (x) - 1}\right ) - \frac {1}{4} \, x^{2} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{4} \, x^{2} \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + \frac {1}{2} \, x {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - \frac {1}{2} \, x {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) - \frac {1}{2} \, {\rm polylog}\left (3, \cosh \relax (x) + \sinh \relax (x)\right ) + \frac {1}{2} \, {\rm polylog}\left (3, -\cosh \relax (x) - \sinh \relax (x)\right ) \]
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 \[ \int x \operatorname {arcoth}\left (e^{x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 62, normalized size = 1.22 \[ \frac {x^{2} \mathrm {arccoth}\left ({\mathrm e}^{x}\right )}{2}-\frac {x^{2} \ln \left ({\mathrm e}^{x}+1\right )}{4}-\frac {x \polylog \left (2, -{\mathrm e}^{x}\right )}{2}+\frac {\polylog \left (3, -{\mathrm e}^{x}\right )}{2}+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{4}+\frac {x \polylog \left (2, {\mathrm e}^{x}\right )}{2}-\frac {\polylog \left (3, {\mathrm e}^{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 59, normalized size = 1.16 \[ \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (e^{x}\right ) - \frac {1}{4} \, x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{4} \, x^{2} \log \left (-e^{x} + 1\right ) - \frac {1}{2} \, x {\rm Li}_2\left (-e^{x}\right ) + \frac {1}{2} \, x {\rm Li}_2\left (e^{x}\right ) + \frac {1}{2} \, {\rm Li}_{3}(-e^{x}) - \frac {1}{2} \, {\rm Li}_{3}(e^{x}) \]
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 \[ \int x\,\mathrm {acoth}\left ({\mathrm {e}}^x\right ) \,d x \]
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 \[ \int x \operatorname {acoth}{\left (e^{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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