Optimal. Leaf size=43 \[ -\frac {1}{2} x \text {Li}_2\left (-e^x\right )+\frac {x \text {Li}_2\left (e^x\right )}{2}+\frac {\text {Li}_3\left (-e^x\right )}{2}-\frac {\text {Li}_3\left (e^x\right )}{2} \]
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Rubi [A] time = 0.04, antiderivative size = 43, 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 = {6213, 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 6213
Rule 6589
Rubi steps
\begin {align*} \int x \tanh ^{-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 {x \text {Li}_2\left (e^x\right )}{2}+\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 {x \text {Li}_2\left (e^x\right )}{2}+\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 {x \text {Li}_2\left (e^x\right )}{2}+\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 = 1.65 \[ \frac {1}{4} \left (-2 x \text {Li}_2\left (-e^x\right )+2 x \text {Li}_2\left (e^x\right )+2 \text {Li}_3\left (-e^x\right )-2 \text {Li}_3\left (e^x\right )+x^2 \log \left (1-e^x\right )-x^2 \log \left (e^x+1\right )+2 x^2 \tanh ^{-1}\left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.54, size = 95, normalized size = 2.21 \[ \frac {1}{4} \, x^{2} \log \left (-\frac {\cosh \relax (x) + \sinh \relax (x) + 1}{\cosh \relax (x) + \sinh \relax (x) - 1}\right ) - \frac {1}{4} \, x^{2} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{4} \, x^{2} \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + \frac {1}{2} \, x {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - \frac {1}{2} \, x {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) - \frac {1}{2} \, {\rm polylog}\left (3, \cosh \relax (x) + \sinh \relax (x)\right ) + \frac {1}{2} \, {\rm polylog}\left (3, -\cosh \relax (x) - \sinh \relax (x)\right ) \]
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 \[ \int x \operatorname {artanh}\left (e^{x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 1.44 \[ \frac {x^{2} \arctanh \left ({\mathrm e}^{x}\right )}{2}-\frac {x^{2} \ln \left ({\mathrm e}^{x}+1\right )}{4}-\frac {x \polylog \left (2, -{\mathrm e}^{x}\right )}{2}+\frac {\polylog \left (3, -{\mathrm e}^{x}\right )}{2}+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{4}+\frac {x \polylog \left (2, {\mathrm e}^{x}\right )}{2}-\frac {\polylog \left (3, {\mathrm e}^{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 59, normalized size = 1.37 \[ \frac {1}{2} \, x^{2} \operatorname {artanh}\left (e^{x}\right ) - \frac {1}{4} \, x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{4} \, x^{2} \log \left (-e^{x} + 1\right ) - \frac {1}{2} \, x {\rm Li}_2\left (-e^{x}\right ) + \frac {1}{2} \, x {\rm Li}_2\left (e^{x}\right ) + \frac {1}{2} \, {\rm Li}_{3}(-e^{x}) - \frac {1}{2} \, {\rm Li}_{3}(e^{x}) \]
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 \[ \int x\,\mathrm {atanh}\left ({\mathrm {e}}^x\right ) \,d x \]
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 \[ \int x \operatorname {atanh}{\left (e^{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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