Integrand size = 200, antiderivative size = 26 \[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x}{4-e^{\frac {1}{-x+x^2}}+\frac {2}{x}+\log (x)} \]
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\[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=\int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left ((-1+x)^2 (4+3 x)-e^{\frac {1}{-x+x^2}} \left (-1+3 x-2 x^2+x^3\right )+(-1+x)^2 x \log (x)\right )}{(1-x)^2 \left (2-\left (-4+e^{\frac {1}{-x+x^2}}\right ) x+x \log (x)\right )^2} \, dx \\ & = \int \left (-\frac {-1+3 x-2 x^2+x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )}-\frac {-2-2 x+13 x^2-4 x^3+x^4-x \log (x)+2 x^2 \log (x)}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx \\ & = -\int \frac {-1+3 x-2 x^2+x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {-2-2 x+13 x^2-4 x^3+x^4-x \log (x)+2 x^2 \log (x)}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx \\ & = -\int \left (\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )}+\frac {2}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )}+\frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)}\right ) \, dx-\int \left (-\frac {2}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}-\frac {2 x}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {13 x^2}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}-\frac {4 x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {x^4}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}-\frac {x \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {2 x^2 \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}\right ) \, dx \\ & = 2 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+2 \int \frac {x}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-2 \int \frac {x^2 \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx+4 \int \frac {x^3}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-13 \int \frac {x^2}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {x^4}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)} \, dx+\int \frac {x \log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx \\ & = 2 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx+2 \int \left (\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx-2 \int \left (\frac {\log (x)}{\left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {2 \log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}\right ) \, dx+4 \int \left (\frac {2}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {3}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx-13 \int \left (\frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {2}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)} \, dx-\int \left (\frac {3}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {4}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {2 x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}+\frac {x^2}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2}\right ) \, dx+\int \left (\frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}+\frac {\log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2}\right ) \, dx \\ & = 2 \left (2 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx\right )+2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-2 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-2 \int \frac {\log (x)}{\left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx-2 \int \frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx-3 \int \frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+4 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-4 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+4 \int \frac {x}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-4 \int \frac {\log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx+8 \int \frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx+12 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-13 \int \frac {1}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-13 \int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-26 \int \frac {1}{(-1+x) \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {x^2}{\left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )^2} \, dx-\int \frac {1}{(-1+x)^2 \left (-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)\right )} \, dx-\int \frac {x}{-2-4 x+e^{\frac {1}{-x+x^2}} x-x \log (x)} \, dx+\int \frac {\log (x)}{(-1+x)^2 \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx+\int \frac {\log (x)}{(-1+x) \left (2+4 x-e^{\frac {1}{-x+x^2}} x+x \log (x)\right )^2} \, dx \\ \end{align*}
Time = 1.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x^2}{2-\left (-4+e^{\frac {1}{(-1+x) x}}\right ) x+x \log (x)} \]
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Time = 1.98 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \(\frac {x^{2}}{x \ln \left (x \right )-{\mathrm e}^{\frac {1}{x \left (-1+x \right )}} x +4 x +2}\) | \(29\) |
risch | \(-\frac {x^{2}}{{\mathrm e}^{\frac {1}{x \left (-1+x \right )}} x -x \ln \left (x \right )-4 x -2}\) | \(30\) |
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Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=-\frac {x^{2}}{x e^{\left (\frac {1}{x^{2} - x}\right )} - x \log \left (x\right ) - 4 \, x - 2} \]
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Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=- \frac {x^{2}}{x e^{\frac {1}{x^{2} - x}} - x \log {\left (x \right )} - 4 x - 2} \]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=-\frac {x^{2} e^{\frac {1}{x}}}{x e^{\left (\frac {1}{x - 1}\right )} - {\left (x \log \left (x\right ) + 4 \, x + 2\right )} e^{\frac {1}{x}}} \]
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Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=-\frac {x^{2}}{x e^{\left (\frac {1}{x^{2} - x}\right )} - x \log \left (x\right ) - 4 \, x - 2} \]
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Time = 13.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 5.88 \[ \int \frac {4 x-5 x^2-2 x^3+3 x^4+e^{\frac {1}{-x+x^2}} \left (x-3 x^2+2 x^3-x^4\right )+\left (x^2-2 x^3+x^4\right ) \log (x)}{4+8 x-12 x^2-16 x^3+16 x^4+e^{\frac {1}{-x+x^2}} \left (-4 x+12 x^3-8 x^4\right )+e^{\frac {2}{-x+x^2}} \left (x^2-2 x^3+x^4\right )+\left (4 x-12 x^3+8 x^4+e^{\frac {1}{-x+x^2}} \left (-2 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)} \, dx=-\frac {x\,{\left (x^3-2\,x^2+x\right )}^2\,\left (-x^4+4\,x^3-13\,x^2+2\,x+2\right )+x\,\ln \left (x\right )\,\left (x-2\,x^2\right )\,{\left (x^3-2\,x^2+x\right )}^2}{{\left (x-1\right )}^2\,\left (4\,x-x\,{\mathrm {e}}^{-\frac {1}{x-x^2}}+x\,\ln \left (x\right )+2\right )\,\left (4\,x^3\,\ln \left (x\right )-x^2\,\ln \left (x\right )-2\,x-5\,x^4\,\ln \left (x\right )+2\,x^5\,\ln \left (x\right )+2\,x^2+15\,x^3-32\,x^4+22\,x^5-6\,x^6+x^7\right )} \]
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