Optimal. Leaf size=27 \[ \frac {\text {Li}_2\left (-e^x\right )}{2}-\frac {\text {Li}_2\left (e^x\right )}{2}+x \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 206
Rule 2245
Rule 2249
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*}
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Mathematica [A] time = 0.04, size = 45, normalized size = 1.67 \[ \frac {\text {Li}_2\left (-e^x\right )}{2}-\frac {\text {Li}_2\left (e^x\right )}{2}-\frac {1}{2} x \log \left (1-e^x\right )+\frac {1}{2} x \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 31, normalized size = 1.15 \[ \frac {1}{2} \, x \log \left (e^{x} + 1\right ) - \frac {1}{2} \, x \log \left (-e^{x} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-e^{x}\right ) - \frac {1}{2} \, {\rm Li}_2\left (e^{x}\right ) \]
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 \[ \int -\frac {x e^{x}}{e^{\left (2 \, x\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 34, normalized size = 1.26 \[ -\frac {x \ln \left (-{\mathrm e}^{x}+1\right )}{2}+\frac {x \ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {\polylog \left (2, -{\mathrm e}^{x}\right )}{2}-\frac {\polylog \left (2, {\mathrm e}^{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 31, normalized size = 1.15 \[ \frac {1}{2} \, x \log \left (e^{x} + 1\right ) - \frac {1}{2} \, x \log \left (-e^{x} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-e^{x}\right ) - \frac {1}{2} \, {\rm Li}_2\left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {x\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1} \,d x \]
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 \[ - \int \frac {x e^{x}}{e^{2 x} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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