Optimal. Leaf size=77 \[ -2 x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (-\frac {e^x}{2}\right )+2 \text {Li}_3\left (-e^x\right )-\text {Li}_3\left (-\frac {e^x}{2}\right )+\frac {x^3}{6}+\frac {1}{2} x^2 \log \left (\frac {e^x}{2}+1\right )-x^2 \log \left (e^x+1\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2263, 2184, 2190, 2531, 2282, 6589} \[ -2 x \text {PolyLog}\left (2,-e^x\right )+x \text {PolyLog}\left (2,-\frac {e^x}{2}\right )+2 \text {PolyLog}\left (3,-e^x\right )-\text {PolyLog}\left (3,-\frac {e^x}{2}\right )+\frac {x^3}{6}+\frac {1}{2} x^2 \log \left (\frac {e^x}{2}+1\right )-x^2 \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2263
Rule 2282
Rule 2531
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2}{2+3 e^x+e^{2 x}} \, dx &=2 \int \frac {x^2}{2+2 e^x} \, dx-2 \int \frac {x^2}{4+2 e^x} \, dx\\ &=\frac {x^3}{6}-2 \int \frac {e^x x^2}{2+2 e^x} \, dx+\int \frac {e^x x^2}{4+2 e^x} \, dx\\ &=\frac {x^3}{6}+\frac {1}{2} x^2 \log \left (1+\frac {e^x}{2}\right )-x^2 \log \left (1+e^x\right )+2 \int x \log \left (1+e^x\right ) \, dx-\int x \log \left (1+\frac {e^x}{2}\right ) \, dx\\ &=\frac {x^3}{6}+\frac {1}{2} x^2 \log \left (1+\frac {e^x}{2}\right )-x^2 \log \left (1+e^x\right )-2 x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (-\frac {e^x}{2}\right )+2 \int \text {Li}_2\left (-e^x\right ) \, dx-\int \text {Li}_2\left (-\frac {e^x}{2}\right ) \, dx\\ &=\frac {x^3}{6}+\frac {1}{2} x^2 \log \left (1+\frac {e^x}{2}\right )-x^2 \log \left (1+e^x\right )-2 x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (-\frac {e^x}{2}\right )+2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx,x,e^x\right )\\ &=\frac {x^3}{6}+\frac {1}{2} x^2 \log \left (1+\frac {e^x}{2}\right )-x^2 \log \left (1+e^x\right )-2 x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (-\frac {e^x}{2}\right )+2 \text {Li}_3\left (-e^x\right )-\text {Li}_3\left (-\frac {e^x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 77, normalized size = 1.00 \[ -x \text {Li}_2\left (-2 e^{-x}\right )+2 x \text {Li}_2\left (-e^{-x}\right )-\text {Li}_3\left (-2 e^{-x}\right )+2 \text {Li}_3\left (-e^{-x}\right )+x^2 \left (-\log \left (e^{-x}+1\right )\right )+\frac {1}{2} x^2 \log \left (2 e^{-x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.42, size = 59, normalized size = 0.77 \[ \frac {1}{6} \, x^{3} - x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x^{2} \log \left (\frac {1}{2} \, e^{x} + 1\right ) + x {\rm Li}_2\left (-\frac {1}{2} \, e^{x}\right ) - 2 \, x {\rm Li}_2\left (-e^{x}\right ) - {\rm polylog}\left (3, -\frac {1}{2} \, e^{x}\right ) + 2 \, {\rm polylog}\left (3, -e^{x}\right ) \]
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 \[ \int \frac {x^{2}}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 62, normalized size = 0.81 \[ \frac {x^{3}}{6}-x^{2} \ln \left ({\mathrm e}^{x}+1\right )+\frac {x^{2} \ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )}{2}-2 x \polylog \left (2, -{\mathrm e}^{x}\right )+x \polylog \left (2, -\frac {{\mathrm e}^{x}}{2}\right )+2 \polylog \left (3, -{\mathrm e}^{x}\right )-\polylog \left (3, -\frac {{\mathrm e}^{x}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 59, normalized size = 0.77 \[ \frac {1}{6} \, x^{3} - x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x^{2} \log \left (\frac {1}{2} \, e^{x} + 1\right ) + x {\rm Li}_2\left (-\frac {1}{2} \, e^{x}\right ) - 2 \, x {\rm Li}_2\left (-e^{x}\right ) - {\rm Li}_{3}(-\frac {1}{2} \, e^{x}) + 2 \, {\rm Li}_{3}(-e^{x}) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+2} \,d x \]
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 \[ \int \frac {x^{2}}{\left (e^{x} + 1\right ) \left (e^{x} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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