Integrand size = 288, antiderivative size = 31 \[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=\frac {e^{x+x^2} x}{-\frac {3}{x}+x \left (x-\log \left (-1+e^4+\log (x)\right )\right )} \]
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\[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=\int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{x+x^2} x \left (-6 \left (1-e^4\right )-3 \left (1-e^4\right ) x-7 \left (1-\frac {6 e^4}{7}\right ) x^2-\left (1-e^4\right ) x^3+\left (1-e^4\right ) x^4+2 \left (1-e^4\right ) x^5+\left (-1+e^4\right ) x^3 (1+2 x) \log \left (-1+e^4+\log (x)\right )-\log (x) \left (-6-3 x-6 x^2-x^3+x^4+2 x^5-x^3 (1+2 x) \log \left (-1+e^4+\log (x)\right )\right )\right )}{\left (1-e^4-\log (x)\right ) \left (3-x^3+x^2 \log \left (-1+e^4+\log (x)\right )\right )^2} \, dx \\ & = \int \left (\frac {e^{x+x^2} x^2 (1+2 x)}{-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )}+\frac {e^{x+x^2} x \left (-6 \left (1-e^4\right )-x^2-\left (1-e^4\right ) x^3+6 \log (x)+x^3 \log (x)\right )}{\left (1-e^4-\log (x)\right ) \left (3-x^3+x^2 \log \left (-1+e^4+\log (x)\right )\right )^2}\right ) \, dx \\ & = \int \frac {e^{x+x^2} x^2 (1+2 x)}{-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )} \, dx+\int \frac {e^{x+x^2} x \left (-6 \left (1-e^4\right )-x^2-\left (1-e^4\right ) x^3+6 \log (x)+x^3 \log (x)\right )}{\left (1-e^4-\log (x)\right ) \left (3-x^3+x^2 \log \left (-1+e^4+\log (x)\right )\right )^2} \, dx \\ & = \int \left (-\frac {6 e^{x+x^2} \left (-1+e^4\right ) x}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2}+\frac {e^{x+x^2} x^3}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2}-\frac {e^{x+x^2} \left (-1+e^4\right ) x^4}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2}-\frac {6 e^{x+x^2} x \log (x)}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2}-\frac {e^{x+x^2} x^4 \log (x)}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2}\right ) \, dx+\int \left (\frac {e^{x+x^2} x^2}{-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )}+\frac {2 e^{x+x^2} x^3}{-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )}\right ) \, dx \\ & = 2 \int \frac {e^{x+x^2} x^3}{-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )} \, dx-6 \int \frac {e^{x+x^2} x \log (x)}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2} \, dx+\left (1-e^4\right ) \int \frac {e^{x+x^2} x^4}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2} \, dx+\left (6 \left (1-e^4\right )\right ) \int \frac {e^{x+x^2} x}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2} \, dx+\int \frac {e^{x+x^2} x^3}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2} \, dx-\int \frac {e^{x+x^2} x^4 \log (x)}{\left (-1+e^4+\log (x)\right ) \left (-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )\right )^2} \, dx+\int \frac {e^{x+x^2} x^2}{-3+x^3-x^2 \log \left (-1+e^4+\log (x)\right )} \, dx \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=-\frac {e^{x+x^2} x^2}{3-x^3+x^2 \log \left (-1+e^4+\log (x)\right )} \]
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Time = 109.76 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {x^{2} {\mathrm e}^{\left (1+x \right ) x}}{x^{3}-x^{2} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-1\right )-3}\) | \(30\) |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{x^{2}+x}}{x^{3}-x^{2} \ln \left (\ln \left (x \right )+{\mathrm e}^{4}-1\right )-3}\) | \(30\) |
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Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=\frac {x^{2} e^{\left (x^{2} + x\right )}}{x^{3} - x^{2} \log \left (e^{4} + \log \left (x\right ) - 1\right ) - 3} \]
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Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=\frac {x^{2} e^{x^{2} + x}}{x^{3} - x^{2} \log {\left (\log {\left (x \right )} - 1 + e^{4} \right )} - 3} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=\frac {x^{2} e^{\left (x^{2} + x\right )}}{x^{3} - x^{2} \log \left (e^{4} + \log \left (x\right ) - 1\right ) - 3} \]
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Time = 0.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=\frac {x^{2} e^{\left (x^{2} + x\right )}}{x^{3} - x^{2} \log \left (e^{4} + \log \left (x\right ) - 1\right ) - 3} \]
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Time = 13.42 (sec) , antiderivative size = 268, normalized size of antiderivative = 8.65 \[ \int \frac {e^{x+x^2} \left (6 x+3 x^2+7 x^3+x^4-x^5-2 x^6+e^4 \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right )\right )+e^{x+x^2} \left (-6 x-3 x^2-6 x^3-x^4+x^5+2 x^6\right ) \log (x)+\left (e^{x+x^2} \left (x^4+2 x^5+e^4 \left (-x^4-2 x^5\right )\right )+e^{x+x^2} \left (-x^4-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )}{-9+6 x^3-x^6+e^4 \left (9-6 x^3+x^6\right )+\left (9-6 x^3+x^6\right ) \log (x)+\left (-6 x^2+2 x^5+e^4 \left (6 x^2-2 x^5\right )+\left (6 x^2-2 x^5\right ) \log (x)\right ) \log \left (-1+e^4+\log (x)\right )+\left (-x^4+e^4 x^4+x^4 \log (x)\right ) \log ^2\left (-1+e^4+\log (x)\right )} \, dx=-\frac {x^3\,\left (6\,{\mathrm {e}}^{x^2+x}-12\,{\mathrm {e}}^{x^2+x+4}+6\,{\mathrm {e}}^{x^2+x+8}-12\,{\mathrm {e}}^{x^2+x}\,\ln \left (x\right )+12\,{\mathrm {e}}^{x^2+x+4}\,\ln \left (x\right )+6\,{\mathrm {e}}^{x^2+x}\,{\ln \left (x\right )}^2\right )-x^5\,\left ({\mathrm {e}}^{x^2+x+4}-{\mathrm {e}}^{x^2+x}+{\mathrm {e}}^{x^2+x}\,\ln \left (x\right )\right )+x^6\,\left ({\mathrm {e}}^{x^2+x}-2\,{\mathrm {e}}^{x^2+x+4}+{\mathrm {e}}^{x^2+x+8}-2\,{\mathrm {e}}^{x^2+x}\,\ln \left (x\right )+2\,{\mathrm {e}}^{x^2+x+4}\,\ln \left (x\right )+{\mathrm {e}}^{x^2+x}\,{\ln \left (x\right )}^2\right )}{\left (x^2\,\ln \left ({\mathrm {e}}^4+\ln \left (x\right )-1\right )-x^3+3\right )\,\left (6\,x+6\,x\,{\ln \left (x\right )}^2-x^3\,\ln \left (x\right )-2\,x^4\,\ln \left (x\right )-12\,x\,{\mathrm {e}}^4+6\,x\,{\mathrm {e}}^8+x^4\,{\ln \left (x\right )}^2-x^3\,{\mathrm {e}}^4-2\,x^4\,{\mathrm {e}}^4+x^4\,{\mathrm {e}}^8-12\,x\,\ln \left (x\right )+x^3+x^4+12\,x\,{\mathrm {e}}^4\,\ln \left (x\right )+2\,x^4\,{\mathrm {e}}^4\,\ln \left (x\right )\right )} \]
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