Optimal. Leaf size=259 \[ -\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}+\frac {4 \text {Li}_3\left (-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}-\frac {4 \text {Li}_3\left (-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {2 x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {2 x^2 \log \left (\frac {2 e^x}{1-\sqrt {5}}+1\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {2 x^2 \log \left (\frac {2 e^x}{1+\sqrt {5}}+1\right )}{\sqrt {5} \left (1+\sqrt {5}\right )} \]
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Rubi [A] time = 0.30, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2263, 2184, 2190, 2531, 2282, 6589} \[ -\frac {4 x \text {PolyLog}\left (2,-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {4 x \text {PolyLog}\left (2,-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}+\frac {4 \text {PolyLog}\left (3,-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}-\frac {4 \text {PolyLog}\left (3,-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}+\frac {2 x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {2 x^2 \log \left (\frac {2 e^x}{1-\sqrt {5}}+1\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {2 x^2 \log \left (\frac {2 e^x}{1+\sqrt {5}}+1\right )}{\sqrt {5} \left (1+\sqrt {5}\right )} \]
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}{-1+e^x+e^{2 x}} \, dx &=\frac {2 \int \frac {x^2}{1-\sqrt {5}+2 e^x} \, dx}{\sqrt {5}}-\frac {2 \int \frac {x^2}{1+\sqrt {5}+2 e^x} \, dx}{\sqrt {5}}\\ &=\frac {2 x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {2 x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {4 \int \frac {e^x x^2}{1-\sqrt {5}+2 e^x} \, dx}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {4 \int \frac {e^x x^2}{1+\sqrt {5}+2 e^x} \, dx}{\sqrt {5} \left (1+\sqrt {5}\right )}\\ &=\frac {2 x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {2 x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}+\frac {4 \int x \log \left (1+\frac {2 e^x}{1-\sqrt {5}}\right ) \, dx}{\sqrt {5} \left (1-\sqrt {5}\right )}-\frac {4 \int x \log \left (1+\frac {2 e^x}{1+\sqrt {5}}\right ) \, dx}{\sqrt {5} \left (1+\sqrt {5}\right )}\\ &=\frac {2 x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {2 x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}-\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}+\frac {4 \int \text {Li}_2\left (-\frac {2 e^x}{1-\sqrt {5}}\right ) \, dx}{\sqrt {5} \left (1-\sqrt {5}\right )}-\frac {4 \int \text {Li}_2\left (-\frac {2 e^x}{1+\sqrt {5}}\right ) \, dx}{\sqrt {5} \left (1+\sqrt {5}\right )}\\ &=\frac {2 x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {2 x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}-\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}+\frac {4 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {2 x}{-1+\sqrt {5}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}-\frac {4 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 x}{1+\sqrt {5}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}\\ &=\frac {2 x^3}{3 \sqrt {5} \left (1-\sqrt {5}\right )}-\frac {2 x^3}{3 \sqrt {5} \left (1+\sqrt {5}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}-\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}+\frac {4 \text {Li}_3\left (-\frac {2 e^x}{1-\sqrt {5}}\right )}{\sqrt {5} \left (1-\sqrt {5}\right )}-\frac {4 \text {Li}_3\left (-\frac {2 e^x}{1+\sqrt {5}}\right )}{\sqrt {5} \left (1+\sqrt {5}\right )}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 172, normalized size = 0.66 \[ \frac {2 \left (-\frac {2 \left (x \text {Li}_2\left (\frac {1}{2} \left (-1+\sqrt {5}\right ) e^{-x}\right )+\text {Li}_3\left (\frac {1}{2} \left (-1+\sqrt {5}\right ) e^{-x}\right )\right )}{\sqrt {5}-1}-\frac {2 \left (x \text {Li}_2\left (-\frac {1}{2} \left (1+\sqrt {5}\right ) e^{-x}\right )+\text {Li}_3\left (-\frac {1}{2} \left (1+\sqrt {5}\right ) e^{-x}\right )\right )}{1+\sqrt {5}}+\frac {x^2 \log \left (1-\frac {1}{2} \left (\sqrt {5}-1\right ) e^{-x}\right )}{\sqrt {5}-1}+\frac {x^2 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right ) e^{-x}+1\right )}{1+\sqrt {5}}\right )}{\sqrt {5}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.42, size = 138, normalized size = 0.53 \[ -\frac {1}{3} \, x^{3} + \frac {1}{5} \, {\left (\sqrt {5} x + 5 \, x\right )} {\rm Li}_2\left (\frac {1}{2} \, {\left (\sqrt {5} + 1\right )} e^{x}\right ) - \frac {1}{5} \, {\left (\sqrt {5} x - 5 \, x\right )} {\rm Li}_2\left (-\frac {1}{2} \, {\left (\sqrt {5} - 1\right )} e^{x}\right ) + \frac {1}{10} \, {\left (\sqrt {5} x^{2} + 5 \, x^{2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {5} + 1\right )} e^{x} + 1\right ) - \frac {1}{10} \, {\left (\sqrt {5} x^{2} - 5 \, x^{2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {5} - 1\right )} e^{x} + 1\right ) - \frac {1}{5} \, {\left (\sqrt {5} + 5\right )} {\rm polylog}\left (3, \frac {1}{2} \, {\left (\sqrt {5} + 1\right )} e^{x}\right ) + \frac {1}{5} \, {\left (\sqrt {5} - 5\right )} {\rm polylog}\left (3, -\frac {1}{2} \, {\left (\sqrt {5} - 1\right )} e^{x}\right ) \]
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 \[ \int \frac {x^{2}}{e^{\left (2 \, x\right )} + e^{x} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\mathrm e}^{x}+{\mathrm e}^{2 x}-1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{e^{\left (2 \, x\right )} + e^{x} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x-1} \,d x \]
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 \[ \int \frac {x^{2}}{e^{2 x} + e^{x} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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