Integrand size = 18, antiderivative size = 293 \[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=-\frac {2 x^3}{3 \sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {2 x^3}{3 \sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {4 x \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {4 x \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {4 \operatorname {PolyLog}\left (3,-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {4 \operatorname {PolyLog}\left (3,-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )} \]
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Time = 0.20 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2295, 2215, 2221, 2611, 2320, 6724} \[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=-\frac {4 x \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {4 x \operatorname {PolyLog}\left (2,-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}+\frac {4 \operatorname {PolyLog}\left (3,-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}-\frac {4 \operatorname {PolyLog}\left (3,-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )}+\frac {2 x^3}{3 \sqrt {3} \left (\sqrt {3}+3 i\right )}-\frac {2 x^3}{3 \sqrt {3} \left (-\sqrt {3}+3 i\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+3 i\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+3 i\right )} \]
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Rule 2215
Rule 2221
Rule 2295
Rule 2320
Rule 2611
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 i) \int \frac {x^2}{3-i \sqrt {3}+2 e^x} \, dx}{\sqrt {3}}+\frac {(2 i) \int \frac {x^2}{3+i \sqrt {3}+2 e^x} \, dx}{\sqrt {3}} \\ & = -\frac {2 x^3}{3 \sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {2 x^3}{3 \sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {(4 i) \int \frac {e^x x^2}{3-i \sqrt {3}+2 e^x} \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}-\frac {(4 i) \int \frac {e^x x^2}{3+i \sqrt {3}+2 e^x} \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )} \\ & = -\frac {2 x^3}{3 \sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {2 x^3}{3 \sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {(4 i) \int x \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right ) \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}+\frac {(4 i) \int x \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right ) \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )} \\ & = -\frac {2 x^3}{3 \sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {2 x^3}{3 \sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {(4 i) \int \text {Li}_2\left (-\frac {2 e^x}{3-i \sqrt {3}}\right ) \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}+\frac {(4 i) \int \text {Li}_2\left (-\frac {2 e^x}{3+i \sqrt {3}}\right ) \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )} \\ & = -\frac {2 x^3}{3 \sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {2 x^3}{3 \sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {(4 i) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 i x}{3 i+\sqrt {3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {3} \left (3-i \sqrt {3}\right )}+\frac {(4 i) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {2 i x}{-3 i+\sqrt {3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {3} \left (3+i \sqrt {3}\right )} \\ & = -\frac {2 x^3}{3 \sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {2 x^3}{3 \sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {4 \text {Li}_3\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {4 \text {Li}_3\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=\frac {2 i \left (\frac {i x^2 \log \left (1+\frac {1}{2} \left (3-i \sqrt {3}\right ) e^{-x}\right )}{3 i+\sqrt {3}}+\frac {i x^2 \log \left (1+\frac {1}{2} \left (3+i \sqrt {3}\right ) e^{-x}\right )}{-3 i+\sqrt {3}}+\frac {2 \left (x \operatorname {PolyLog}\left (2,-\frac {1}{2} i \left (-3 i+\sqrt {3}\right ) e^{-x}\right )+\operatorname {PolyLog}\left (3,-\frac {1}{2} i \left (-3 i+\sqrt {3}\right ) e^{-x}\right )\right )}{3+i \sqrt {3}}-\frac {2 i \left (x \operatorname {PolyLog}\left (2,\frac {1}{2} i \left (3 i+\sqrt {3}\right ) e^{-x}\right )+\operatorname {PolyLog}\left (3,\frac {1}{2} i \left (3 i+\sqrt {3}\right ) e^{-x}\right )\right )}{3 i+\sqrt {3}}\right )}{\sqrt {3}} \]
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\[\int \frac {x^{2}}{3+3 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}}d x\]
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Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.51 \[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=\frac {1}{9} \, x^{3} - \frac {1}{3} \, {\left (-i \, \sqrt {3} x + x\right )} {\rm Li}_2\left (-\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x}\right ) - \frac {1}{3} \, {\left (i \, \sqrt {3} x + x\right )} {\rm Li}_2\left (-\frac {1}{6} \, {\left (-i \, \sqrt {3} + 3\right )} e^{x}\right ) - \frac {1}{6} \, {\left (-i \, \sqrt {3} x^{2} + x^{2}\right )} \log \left (\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x} + 1\right ) - \frac {1}{6} \, {\left (i \, \sqrt {3} x^{2} + x^{2}\right )} \log \left (\frac {1}{6} \, {\left (-i \, \sqrt {3} + 3\right )} e^{x} + 1\right ) - \frac {1}{3} \, {\left (-i \, \sqrt {3} - 1\right )} {\rm polylog}\left (3, \frac {1}{6} \, {\left (i \, \sqrt {3} - 3\right )} e^{x}\right ) - \frac {1}{3} \, {\left (i \, \sqrt {3} - 1\right )} {\rm polylog}\left (3, \frac {1}{6} \, {\left (-i \, \sqrt {3} - 3\right )} e^{x}\right ) \]
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\[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=\int \frac {x^{2}}{e^{2 x} + 3 e^{x} + 3}\, dx \]
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\[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=\int { \frac {x^{2}}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3} \,d x } \]
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\[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=\int { \frac {x^{2}}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3} \,d x } \]
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Timed out. \[ \int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx=\int \frac {x^2}{{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+3} \,d x \]
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