Optimal. Leaf size=44 \[ -\text {Li}_2\left (-e^x\right )+\frac {x^2}{2}+\frac {x}{e^x+1}-x-x \log \left (e^x+1\right )+\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6688, 2185, 2184, 2190, 2279, 2391, 2191, 2282, 36, 29, 31} \[ -\text {PolyLog}\left (2,-e^x\right )+\frac {x^2}{2}+\frac {x}{e^x+1}-x-x \log \left (e^x+1\right )+\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 6688
Rubi steps
\begin {align*} \int \frac {x}{1+2 e^x+e^{2 x}} \, dx &=\int \frac {x}{\left (1+e^x\right )^2} \, dx\\ &=-\int \frac {e^x x}{\left (1+e^x\right )^2} \, dx+\int \frac {x}{1+e^x} \, dx\\ &=\frac {x}{1+e^x}+\frac {x^2}{2}-\int \frac {1}{1+e^x} \, dx-\int \frac {e^x x}{1+e^x} \, dx\\ &=\frac {x}{1+e^x}+\frac {x^2}{2}-x \log \left (1+e^x\right )+\int \log \left (1+e^x\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right )\\ &=\frac {x}{1+e^x}+\frac {x^2}{2}-x \log \left (1+e^x\right )-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right )\\ &=-x+\frac {x}{1+e^x}+\frac {x^2}{2}+\log \left (1+e^x\right )-x \log \left (1+e^x\right )-\text {Li}_2\left (-e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 38, normalized size = 0.86 \[ -\text {Li}_2\left (-e^x\right )+\frac {1}{2} x \left (x+\frac {2}{e^x+1}-2\right )-(x-1) \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 49, normalized size = 1.11 \[ \frac {x^{2} - 2 \, {\left (e^{x} + 1\right )} {\rm Li}_2\left (-e^{x}\right ) + {\left (x^{2} - 2 \, x\right )} e^{x} - 2 \, {\left ({\left (x - 1\right )} e^{x} + x - 1\right )} \log \left (e^{x} + 1\right )}{2 \, {\left (e^{x} + 1\right )}} \]
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 \[ \int \frac {x}{e^{\left (2 \, x\right )} + 2 \, e^{x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 38, normalized size = 0.86 \[ \frac {x^{2}}{2}-\frac {x \,{\mathrm e}^{x}}{{\mathrm e}^{x}+1}-x \ln \left ({\mathrm e}^{x}+1\right )-\dilog \left ({\mathrm e}^{x}+1\right )+\ln \left ({\mathrm e}^{x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 37, normalized size = 0.84 \[ \frac {1}{2} \, x^{2} - x \log \left (e^{x} + 1\right ) - x + \frac {x}{e^{x} + 1} - {\rm Li}_2\left (-e^{x}\right ) + \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1} \,d x \]
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 \[ \frac {x}{e^{x} + 1} + \int \frac {x - 1}{e^{x} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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