Integrand size = 205, antiderivative size = 32 \[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=\log \left (\left (4+\frac {-1-e^{1-x}+2 x}{\log (x)}\right ) \log \left (-x+\log ^2(x)\right )\right ) \]
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\[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=\int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )\right )}{x \log (x) \left (e+e^x-2 e^x x-4 e^x \log (x)\right ) \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx \\ & = \int \left (\frac {e^x \left (4+x+2 x^2+4 x \log (x)\right )}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )}-\frac {-x \log (x)+2 \log ^2(x)+x \log \left (-x+\log ^2(x)\right )+x^2 \log (x) \log \left (-x+\log ^2(x)\right )-\log ^2(x) \log \left (-x+\log ^2(x)\right )-x \log ^3(x) \log \left (-x+\log ^2(x)\right )}{x \log (x) \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )}\right ) \, dx \\ & = \int \frac {e^x \left (4+x+2 x^2+4 x \log (x)\right )}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx-\int \frac {-x \log (x)+2 \log ^2(x)+x \log \left (-x+\log ^2(x)\right )+x^2 \log (x) \log \left (-x+\log ^2(x)\right )-\log ^2(x) \log \left (-x+\log ^2(x)\right )-x \log ^3(x) \log \left (-x+\log ^2(x)\right )}{x \log (x) \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx \\ & = \int \left (\frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)}+\frac {4 e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )}+\frac {2 e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)}+\frac {4 e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)}\right ) \, dx-\int \left (\frac {1+x \log (x)}{x \log (x)}+\frac {-x+2 \log (x)}{x \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )}\right ) \, dx \\ & = 2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\int \frac {1+x \log (x)}{x \log (x)} \, dx-\int \frac {-x+2 \log (x)}{x \left (x-\log ^2(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx \\ & = \log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\int \left (1+\frac {1}{x \log (x)}\right ) \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx \\ & = -x+\log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx \\ & = -x+\log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = -x-\log (\log (x))+\log \left (\log \left (-x+\log ^2(x)\right )\right )+2 \int \frac {e^x x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+4 \int \frac {e^x}{x \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )} \, dx+4 \int \frac {e^x \log (x)}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx+\int \frac {e^x}{-e-e^x+2 e^x x+4 e^x \log (x)} \, dx \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=-x-\log (\log (x))+\log \left (-e-e^x+2 e^x x+4 e^x \log (x)\right )+\log \left (\log \left (-x+\log ^2(x)\right )\right ) \]
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Time = 13.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\ln \left (\frac {x}{2}-\frac {{\mathrm e}^{1-x}}{4}+\ln \left (x \right )-\frac {1}{4}\right )-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (\ln \left (x \right )^{2}-x \right )\right )\) | \(33\) |
parallelrisch | \(-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (\ln \left (x \right )^{2}-x \right )\right )+\ln \left (x +2 \ln \left (x \right )-\frac {{\mathrm e}^{1-x}}{2}-\frac {1}{2}\right )\) | \(33\) |
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=\log \left (2 \, x - e^{\left (-x + 1\right )} + 4 \, \log \left (x\right ) - 1\right ) + \log \left (\log \left (\log \left (x\right )^{2} - x\right )\right ) - \log \left (\log \left (x\right )\right ) \]
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Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=\log {\left (- 2 x + e^{1 - x} - 4 \log {\left (x \right )} + 1 \right )} - \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (- x + \log {\left (x \right )}^{2} \right )} \right )} \]
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=-x + \log \left (\frac {1}{2} \, x + \log \left (x\right ) - \frac {1}{4}\right ) + \log \left (\frac {{\left (2 \, x + 4 \, \log \left (x\right ) - 1\right )} e^{x} - e}{2 \, x + 4 \, \log \left (x\right ) - 1}\right ) + \log \left (\log \left (\log \left (x\right )^{2} - x\right )\right ) - \log \left (\log \left (x\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=\log \left (2 \, x - e^{\left (-x + 1\right )} + 4 \, \log \left (x\right ) - 1\right ) + \log \left (\log \left (\log \left (x\right )^{2} - x\right )\right ) - \log \left (\log \left (x\right )\right ) \]
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Time = 8.48 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {\left (x+e^{1-x} x-2 x^2\right ) \log (x)+\left (-2-2 e^{1-x}\right ) \log ^2(x)+8 \log ^3(x)+\left (-x-e^{1-x} x+2 x^2+\left (-2 x^2-e^{1-x} x^2\right ) \log (x)+\left (1+e^{1-x}-2 x\right ) \log ^2(x)+\left (2 x+e^{1-x} x\right ) \log ^3(x)\right ) \log \left (-x+\log ^2(x)\right )}{\left (\left (x^2+e^{1-x} x^2-2 x^3\right ) \log (x)-4 x^2 \log ^2(x)+\left (-x-e^{1-x} x+2 x^2\right ) \log ^3(x)+4 x \log ^4(x)\right ) \log \left (-x+\log ^2(x)\right )} \, dx=\ln \left (\frac {2\,x-{\mathrm {e}}^{1-x}+4\,\ln \left (x\right )-1}{x}\right )+\ln \left (\frac {2\,x+x\,{\mathrm {e}}^{1-x}+4}{x}\right )+\ln \left (\ln \left ({\ln \left (x\right )}^2-x\right )\right )-\ln \left (\frac {4\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+x\,{\mathrm {e}}^{1-x}\,\ln \left (x\right )}{x}\right )+\ln \left (x\right ) \]
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