Integrand size = 118, antiderivative size = 25 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} x^2 \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right ) \]
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Time = 3.56 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6820, 12, 14, 6874, 2631, 6819} \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} x^2 \log ^2\left (x-e^{e^{e^{x^2+x}}}\right ) \]
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Rule 12
Rule 14
Rule 2631
Rule 6819
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2}{27} x \log \left (-e^{e^{e^{x+x^2}}}+x\right ) \left (\frac {x \left (-1+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} (1+2 x)\right )}{e^{e^{e^{x+x^2}}}-x}+\log \left (-e^{e^{e^{x+x^2}}}+x\right )\right ) \, dx \\ & = \frac {2}{27} \int x \log \left (-e^{e^{e^{x+x^2}}}+x\right ) \left (\frac {x \left (-1+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} (1+2 x)\right )}{e^{e^{e^{x+x^2}}}-x}+\log \left (-e^{e^{e^{x+x^2}}}+x\right )\right ) \, dx \\ & = \frac {1}{27} x^2 \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right ) \\ \end{align*}
\[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx \]
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Time = 13.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {\ln \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\left (1+x \right ) x}}}+x \right )^{2} x^{2}}{27}\) | \(21\) |
parallelrisch | \(\frac {\ln \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+x}}}+x \right )^{2} x^{2}}{27}\) | \(21\) |
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left ({\left (x e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )}\right )} - e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )} + e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )} e^{\left (-x^{2} - x - e^{\left (x^{2} + x\right )}\right )}\right )^{2} \]
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Time = 3.81 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {x^{2} \log {\left (x - e^{e^{e^{x^{2} + x}}} \right )}^{2}}{27} \]
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none
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left (x - e^{\left (e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.62 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {1}{27} \, x^{2} \log \left ({\left (x e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )}\right )} - e^{\left (x^{2} + x + e^{\left (x^{2} + x\right )} + e^{\left (e^{\left (x^{2} + x\right )}\right )}\right )}\right )} e^{\left (-x^{2} - x - e^{\left (x^{2} + x\right )}\right )}\right )^{2} \]
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Time = 8.76 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-2 x^2+e^{e^{e^{x+x^2}}+e^{x+x^2}+x+x^2} \left (2 x^2+4 x^3\right )\right ) \log \left (-e^{e^{e^{x+x^2}}}+x\right )+\left (2 e^{e^{e^{x+x^2}}} x-2 x^2\right ) \log ^2\left (-e^{e^{e^{x+x^2}}}+x\right )}{27 e^{e^{e^{x+x^2}}}-27 x} \, dx=\frac {x^2\,{\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}}\right )}^2}{27} \]
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