Optimal. Leaf size=27 \[ \frac{1}{2} \text{PolyLog}\left (2,-e^x\right )-\frac{1}{2} \text{PolyLog}\left (2,e^x\right )+x \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0581555, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {2249, 206, 2245, 2282, 5912} \[ \frac{1}{2} \text{PolyLog}\left (2,-e^x\right )-\frac{1}{2} \text{PolyLog}\left (2,e^x\right )+x \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2249
Rule 206
Rule 2245
Rule 2282
Rule 5912
Rubi steps
\begin{align*} \int \frac{e^x x}{1-e^{2 x}} \, dx &=x \tanh ^{-1}\left (e^x\right )-\int \tanh ^{-1}\left (e^x\right ) \, dx\\ &=x \tanh ^{-1}\left (e^x\right )-\operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{x} \, dx,x,e^x\right )\\ &=x \tanh ^{-1}\left (e^x\right )+\frac{\text{Li}_2\left (-e^x\right )}{2}-\frac{\text{Li}_2\left (e^x\right )}{2}\\ \end{align*}
Mathematica [A] time = 0.0323428, size = 45, normalized size = 1.67 \[ \frac{1}{2} \text{PolyLog}\left (2,-e^x\right )-\frac{1}{2} \text{PolyLog}\left (2,e^x\right )-\frac{1}{2} x \log \left (1-e^x\right )+\frac{1}{2} x \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 34, normalized size = 1.3 \begin{align*}{\frac{x\ln \left ( 1+{{\rm e}^{x}} \right ) }{2}}+{\frac{{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) }{2}}-{\frac{x\ln \left ( 1-{{\rm e}^{x}} \right ) }{2}}-{\frac{{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21072, size = 42, normalized size = 1.56 \begin{align*} \frac{1}{2} \, x \log \left (e^{x} + 1\right ) - \frac{1}{2} \, x \log \left (-e^{x} + 1\right ) + \frac{1}{2} \,{\rm Li}_2\left (-e^{x}\right ) - \frac{1}{2} \,{\rm Li}_2\left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52436, size = 104, normalized size = 3.85 \begin{align*} \frac{1}{2} \, x \log \left (e^{x} + 1\right ) - \frac{1}{2} \, x \log \left (-e^{x} + 1\right ) + \frac{1}{2} \,{\rm Li}_2\left (-e^{x}\right ) - \frac{1}{2} \,{\rm Li}_2\left (e^{x}\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 \frac{x e^{x}}{e^{2 x} - 1}\, 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 -\frac{x e^{x}}{e^{\left (2 \, x\right )} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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