Optimal. Leaf size=54 \[ \frac {x^2}{4}+\frac {1}{2} x \log \left (1+\frac {e^x}{2}\right )-x \log \left (1+e^x\right )-\text {Li}_2\left (-e^x\right )+\frac {1}{2} \text {Li}_2\left (-\frac {e^x}{2}\right ) \]
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Rubi [A]
time = 0.08, 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 = {2295, 2215,
2221, 2317, 2438} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2295
Rule 2317
Rule 2438
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} \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,e^x\right )+\text {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 = 0.05, size = 49, normalized size = 0.91 \begin {gather*} -x \log \left (1+e^{-x}\right )+\frac {1}{2} x \log \left (1+2 e^{-x}\right )-\frac {1}{2} \text {Li}_2\left (-2 e^{-x}\right )+\text {Li}_2\left (-e^{-x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 41, normalized size = 0.76
method | result | size |
default | \(\frac {x^{2}}{4}+\frac {x \ln \left (1+\frac {{\mathrm e}^{x}}{2}\right )}{2}-x \ln \left (1+{\mathrm e}^{x}\right )-\polylog \left (2, -{\mathrm e}^{x}\right )+\frac {\polylog \left (2, -\frac {{\mathrm e}^{x}}{2}\right )}{2}\) | \(41\) |
risch | \(\frac {x^{2}}{4}+\frac {x \ln \left (1+\frac {{\mathrm e}^{x}}{2}\right )}{2}-x \ln \left (1+{\mathrm e}^{x}\right )-\polylog \left (2, -{\mathrm e}^{x}\right )+\frac {\polylog \left (2, -\frac {{\mathrm e}^{x}}{2}\right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 38, normalized size = 0.70 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 38, normalized size = 0.70 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \int \frac {x}{\left (e^{x} + 1\right ) \left (e^{x} + 2\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x}{{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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