Optimal. Leaf size=54 \[ -\text {Li}_2\left (-e^x\right )+\frac {1}{2} \text {Li}_2\left (-\frac {e^x}{2}\right )+\frac {x^2}{4}+\frac {1}{2} x \log \left (\frac {e^x}{2}+1\right )-x \log \left (e^x+1\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2263, 2184, 2190, 2279, 2391} \[ -\text {PolyLog}\left (2,-e^x\right )+\frac {1}{2} \text {PolyLog}\left (2,-\frac {e^x}{2}\right )+\frac {x^2}{4}+\frac {1}{2} x \log \left (\frac {e^x}{2}+1\right )-x \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2263
Rule 2279
Rule 2391
Rubi steps
\begin {align*} \int \frac {x}{2+3 e^x+e^{2 x}} \, dx &=2 \int \frac {x}{2+2 e^x} \, dx-2 \int \frac {x}{4+2 e^x} \, dx\\ &=\frac {x^2}{4}-2 \int \frac {e^x x}{2+2 e^x} \, dx+\int \frac {e^x x}{4+2 e^x} \, dx\\ &=\frac {x^2}{4}+\frac {1}{2} x \log \left (1+\frac {e^x}{2}\right )-x \log \left (1+e^x\right )-\frac {1}{2} \int \log \left (1+\frac {e^x}{2}\right ) \, dx+\int \log \left (1+e^x\right ) \, dx\\ &=\frac {x^2}{4}+\frac {1}{2} x \log \left (1+\frac {e^x}{2}\right )-x \log \left (1+e^x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right )\\ &=\frac {x^2}{4}+\frac {1}{2} x \log \left (1+\frac {e^x}{2}\right )-x \log \left (1+e^x\right )-\text {Li}_2\left (-e^x\right )+\frac {1}{2} \text {Li}_2\left (-\frac {e^x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 49, normalized size = 0.91 \[ -\frac {1}{2} \text {Li}_2\left (-2 e^{-x}\right )+\text {Li}_2\left (-e^{-x}\right )-x \log \left (e^{-x}+1\right )+\frac {1}{2} x \log \left (2 e^{-x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 38, normalized size = 0.70 \[ \frac {1}{4} \, x^{2} - x \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x \log \left (\frac {1}{2} \, e^{x} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-\frac {1}{2} \, e^{x}\right ) - {\rm Li}_2\left (-e^{x}\right ) \]
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 \[ \int \frac {x}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 41, normalized size = 0.76 \[ \frac {x^{2}}{4}-x \ln \left ({\mathrm e}^{x}+1\right )+\frac {x \ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )}{2}-\polylog \left (2, -{\mathrm e}^{x}\right )+\frac {\polylog \left (2, -\frac {{\mathrm e}^{x}}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 38, normalized size = 0.70 \[ \frac {1}{4} \, x^{2} - x \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x \log \left (\frac {1}{2} \, e^{x} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-\frac {1}{2} \, e^{x}\right ) - {\rm Li}_2\left (-e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+2} \,d x \]
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 \[ \int \frac {x}{\left (e^{x} + 1\right ) \left (e^{x} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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