Integrand size = 82, antiderivative size = 23 \[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=\frac {x}{-\frac {9}{4}-e^{x+x^2 \log (3 x)}+x} \]
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\[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=\int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx \\ & = \int \left (-\frac {36}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}-\frac {16 e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {16 e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {16\ 3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}+\frac {32\ 3^{x^2} e^x x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2}\right ) \, dx \\ & = -\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+32 \int \frac {3^{x^2} e^x x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx \\ & = -\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+32 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2} \log (3 x)}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx \\ & = -\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-32 \int \frac {\int \frac {3^{x^2} e^x x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx}{x} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+(32 \log (3 x)) \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx \\ & = -\left (16 \int \frac {e^{x+x^2 \log (3 x)}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx\right )+16 \int \frac {e^{x+x^2 \log (3 x)} x}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+16 \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx-32 \int \frac {\int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx}{x} \, dx-36 \int \frac {1}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx+(32 \log (3 x)) \int \frac {e^{x+x^2 \log (3)} x^{2+x^2}}{\left (9-4 x+4\ 3^{x^2} e^x x^{x^2}\right )^2} \, dx \\ \end{align*}
\[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=\int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx \]
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Time = 0.85 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {4 x}{4 x -4 \left (3 x \right )^{x^{2}} {\mathrm e}^{x}-9}\) | \(22\) |
parallelrisch | \(\frac {4 x}{4 x -4 \,{\mathrm e}^{x^{2} \ln \left (3 x \right )+x}-9}\) | \(24\) |
norman | \(\frac {4 \,{\mathrm e}^{x^{2} \ln \left (3 x \right )+x}+9}{4 x -4 \,{\mathrm e}^{x^{2} \ln \left (3 x \right )+x}-9}\) | \(37\) |
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=\frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \left (3 \, x\right ) + x\right )} - 9} \]
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Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=- \frac {x}{- x + e^{x^{2} \log {\left (3 x \right )} + x} + \frac {9}{4}} \]
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Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=\frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \left (3\right ) + x^{2} \log \left (x\right ) + x\right )} - 9} \]
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Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=\frac {4 \, x}{4 \, x - 4 \, e^{\left (x^{2} \log \left (3 \, x\right ) + x\right )} - 9} \]
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Time = 11.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int \frac {-36+e^{x+x^2 \log (3 x)} \left (-16+16 x+16 x^2+32 x^2 \log (3 x)\right )}{81+16 e^{2 x+2 x^2 \log (3 x)}+e^{x+x^2 \log (3 x)} (72-32 x)-72 x+16 x^2} \, dx=-\frac {52\,x+72\,x^2\,\ln \left (3\,x\right )-32\,x^3\,\ln \left (3\,x\right )+20\,x^2-16\,x^3}{\left (4\,{\mathrm {e}}^x\,{\left (3\,x\right )}^{x^2}-4\,x+9\right )\,\left (5\,x+18\,x\,\ln \left (3\,x\right )-8\,x^2\,\ln \left (3\,x\right )-4\,x^2+13\right )} \]
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