Integrand size = 140, antiderivative size = 25 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \]
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\[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=\int \frac {\exp \left (\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right ) \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8 \exp \left (\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right ) x \left (-1+2 x^2\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )}+\frac {4 \exp \left (x^2+\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right ) x \left (-1+\log \left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )}+\frac {\exp \left (\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right ) \left (-4+8 x^2+3 x^2 \log ^2\left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx \\ & = 4 \int \frac {\exp \left (x^2+\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right ) x \left (-1+\log \left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx-8 \int \frac {\exp \left (\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right ) x \left (-1+2 x^2\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx+\int \frac {\exp \left (\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}\right ) \left (-4+8 x^2+3 x^2 \log ^2\left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x \left (-1+\log \left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx-8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x \left (-1+2 x^2\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx+\int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \left (-4+8 x^2+3 x^2 \log ^2\left (e^{x^2}+2 x\right )\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 4 \int \left (-\frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )}+\frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )}\right ) \, dx-8 \int \left (-\frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )}+\frac {2 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx+\int \left (3 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2+\frac {4 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \left (-1+2 x^2\right )}{\log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx \\ & = 3 \int e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2 \, dx-4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \left (-1+2 x^2\right )}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx-16 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 3 \int e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2 \, dx+4 \int \left (-\frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}}}{\log ^2\left (e^{x^2}+2 x\right )}+\frac {2 e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2}{\log ^2\left (e^{x^2}+2 x\right )}\right ) \, dx-4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx-16 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx \\ & = 3 \int e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2 \, dx-4 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}}}{\log ^2\left (e^{x^2}+2 x\right )} \, dx-4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+4 \int \frac {e^{x^2+x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\log \left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^2}{\log ^2\left (e^{x^2}+2 x\right )} \, dx+8 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx-16 \int \frac {e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} x^3}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{x^3+\frac {2 e^{x^2}}{\log \left (e^{x^2}+2 x\right )}} \]
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Time = 3.96 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
risch | \({\mathrm e}^{\frac {x^{3} \ln \left ({\mathrm e}^{x^{2}}+2 x \right )+2 \,{\mathrm e}^{x^{2}}}{\ln \left ({\mathrm e}^{x^{2}}+2 x \right )}}\) | \(34\) |
parallelrisch | \({\mathrm e}^{\frac {x^{3} \ln \left ({\mathrm e}^{x^{2}}+2 x \right )+2 \,{\mathrm e}^{x^{2}}}{\ln \left ({\mathrm e}^{x^{2}}+2 x \right )}}\) | \(34\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{\left (\frac {x^{3} \log \left (2 \, x + e^{\left (x^{2}\right )}\right ) + 2 \, e^{\left (x^{2}\right )}}{\log \left (2 \, x + e^{\left (x^{2}\right )}\right )}\right )} \]
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Timed out. \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx=e^{\left (x^{3} + \frac {2 \, e^{\left (x^{2}\right )}}{\log \left (2 \, x + e^{\left (x^{2}\right )}\right )}\right )} \]
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Time = 8.93 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {2 e^{x^2}+x^3 \log \left (e^{x^2}+2 x\right )}{\log \left (e^{x^2}+2 x\right )}} \left (-4 e^{x^2}-4 e^{2 x^2} x+\left (4 e^{2 x^2} x+8 e^{x^2} x^2\right ) \log \left (e^{x^2}+2 x\right )+\left (3 e^{x^2} x^2+6 x^3\right ) \log ^2\left (e^{x^2}+2 x\right )\right )}{\left (e^{x^2}+2 x\right ) \log ^2\left (e^{x^2}+2 x\right )} \, dx={\mathrm {e}}^{x^3}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{x^2}}{\ln \left (2\,x+{\mathrm {e}}^{x^2}\right )}} \]
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