Integrand size = 112, antiderivative size = 32 \[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=\frac {x \left (5+\frac {1}{4} e^{e^{-x} x}+x-\log \left (2 x^2\right )\right )}{x+\log (x)} \]
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\[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=\int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )\right )}{4 (x+\log (x))^2} \, dx \\ & = \frac {1}{4} \int \frac {e^{-x} \left (e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )\right )}{(x+\log (x))^2} \, dx \\ & = \frac {1}{4} \int \left (-\frac {e^{-x+e^{-x} x} \left (e^x-x^2+x^3-e^x \log (x)-x \log (x)+x^2 \log (x)\right )}{(x+\log (x))^2}+\frac {4 \left (-5-3 x+x^2+3 \log (x)+2 x \log (x)+\log \left (2 x^2\right )-\log (x) \log \left (2 x^2\right )\right )}{(x+\log (x))^2}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{-x+e^{-x} x} \left (e^x-x^2+x^3-e^x \log (x)-x \log (x)+x^2 \log (x)\right )}{(x+\log (x))^2} \, dx\right )+\int \frac {-5-3 x+x^2+3 \log (x)+2 x \log (x)+\log \left (2 x^2\right )-\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} \left (e^x-x^2+x^3-e^x \log (x)-x \log (x)+x^2 \log (x)\right )}{(x+\log (x))^2} \, dx\right )+\int \left (-\frac {5}{(x+\log (x))^2}-\frac {3 x}{(x+\log (x))^2}+\frac {x^2}{(x+\log (x))^2}+\frac {3 \log (x)}{(x+\log (x))^2}+\frac {2 x \log (x)}{(x+\log (x))^2}-\frac {(-1+\log (x)) \log \left (2 x^2\right )}{(x+\log (x))^2}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \left (-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2}-\frac {e^{x-e^{-x} \left (-1+e^x\right ) x} (-1+\log (x))}{(x+\log (x))^2}-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x \log (x)}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2 \log (x)}{(x+\log (x))^2}\right ) \, dx\right )+2 \int \frac {x \log (x)}{(x+\log (x))^2} \, dx-3 \int \frac {x}{(x+\log (x))^2} \, dx+3 \int \frac {\log (x)}{(x+\log (x))^2} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx-\int \frac {(-1+\log (x)) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx \\ & = \frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{x-e^{-x} \left (-1+e^x\right ) x} (-1+\log (x))}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x \log (x)}{(x+\log (x))^2} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2 \log (x)}{(x+\log (x))^2} \, dx+2 \int \left (-\frac {x^2}{(x+\log (x))^2}+\frac {x}{x+\log (x)}\right ) \, dx-3 \int \frac {x}{(x+\log (x))^2} \, dx+3 \int \left (-\frac {x}{(x+\log (x))^2}+\frac {1}{x+\log (x)}\right ) \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx-\int \left (-\frac {\log \left (2 x^2\right )}{(x+\log (x))^2}+\frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2}\right ) \, dx \\ & = \frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{e^{-x} x} (-1+\log (x))}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \left (-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)}\right ) \, dx-\frac {1}{4} \int \left (-\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^3}{(x+\log (x))^2}+\frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)}\right ) \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx \\ & = \frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx+\frac {1}{4} \int \left (\frac {e^{e^{-x} x} (-1-x)}{(x+\log (x))^2}+\frac {e^{e^{-x} x}}{x+\log (x)}\right ) \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx \\ & = \frac {1}{4} \int \frac {e^{e^{-x} x} (-1-x)}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{e^{-x} x}}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx \\ & = \frac {1}{4} \int \frac {e^{e^{-x} x}}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx+\frac {1}{4} \int \left (-\frac {e^{e^{-x} x}}{(x+\log (x))^2}-\frac {e^{e^{-x} x} x}{(x+\log (x))^2}\right ) \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{e^{-x} x}}{(x+\log (x))^2} \, dx\right )-\frac {1}{4} \int \frac {e^{e^{-x} x} x}{(x+\log (x))^2} \, dx+\frac {1}{4} \int \frac {e^{e^{-x} x}}{x+\log (x)} \, dx+\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x}{x+\log (x)} \, dx-\frac {1}{4} \int \frac {e^{-e^{-x} \left (-1+e^x\right ) x} x^2}{x+\log (x)} \, dx-2 \int \frac {x^2}{(x+\log (x))^2} \, dx+2 \int \frac {x}{x+\log (x)} \, dx-2 \left (3 \int \frac {x}{(x+\log (x))^2} \, dx\right )+3 \int \frac {1}{x+\log (x)} \, dx-5 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x^2}{(x+\log (x))^2} \, dx+\int \frac {\log \left (2 x^2\right )}{(x+\log (x))^2} \, dx-\int \frac {\log (x) \log \left (2 x^2\right )}{(x+\log (x))^2} \, dx \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=\frac {x \left (e^{e^{-x} x}+4 (5+x)-4 \log \left (2 x^2\right )\right )}{4 (x+\log (x))} \]
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Time = 5.40 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56
method | result | size |
parallelrisch | \(-\frac {-240 x +48 x \ln \left (2 x^{2}\right )+48 x \ln \left (x \right )-48 \ln \left (x \right ) \ln \left ({\mathrm e}^{x}\right )-12 \,{\mathrm e}^{x \,{\mathrm e}^{-x}} x -48 x \ln \left ({\mathrm e}^{x}\right )}{48 \left (x +\ln \left (x \right )\right )}\) | \(50\) |
risch | \(-2 x +\frac {\left (10+i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 \ln \left (2\right )+6 x \right ) x}{2 x +2 \ln \left (x \right )}+\frac {{\mathrm e}^{x \,{\mathrm e}^{-x}} x}{4 x +4 \ln \left (x \right )}\) | \(88\) |
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=\frac {4 \, x^{2} + x e^{\left (x e^{\left (-x\right )}\right )} - 4 \, x \log \left (2\right ) - 8 \, x \log \left (x\right ) + 20 \, x}{4 \, {\left (x + \log \left (x\right )\right )}} \]
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Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=- 2 x + \frac {x e^{x e^{- x}}}{4 x + 4 \log {\left (x \right )}} + \frac {3 x^{2} - x \log {\left (2 \right )} + 5 x}{x + \log {\left (x \right )}} \]
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\[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=\int { \frac {4 \, {\left (2 \, x + 3\right )} e^{x} \log \left (x\right ) - {\left (x^{3} - x^{2} + {\left (x^{2} - x - e^{x}\right )} \log \left (x\right ) + e^{x}\right )} e^{\left (x e^{\left (-x\right )}\right )} + 4 \, {\left (x^{2} - 3 \, x - 5\right )} e^{x} - 4 \, {\left (e^{x} \log \left (x\right ) - e^{x}\right )} \log \left (2 \, x^{2}\right )}{4 \, {\left (x^{2} e^{x} + 2 \, x e^{x} \log \left (x\right ) + e^{x} \log \left (x\right )^{2}\right )}} \,d x } \]
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\[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=\int { \frac {4 \, {\left (2 \, x + 3\right )} e^{x} \log \left (x\right ) - {\left (x^{3} - x^{2} + {\left (x^{2} - x - e^{x}\right )} \log \left (x\right ) + e^{x}\right )} e^{\left (x e^{\left (-x\right )}\right )} + 4 \, {\left (x^{2} - 3 \, x - 5\right )} e^{x} - 4 \, {\left (e^{x} \log \left (x\right ) - e^{x}\right )} \log \left (2 \, x^{2}\right )}{4 \, {\left (x^{2} e^{x} + 2 \, x e^{x} \log \left (x\right ) + e^{x} \log \left (x\right )^{2}\right )}} \,d x } \]
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Time = 13.94 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.28 \[ \int \frac {e^x \left (-20-12 x+4 x^2\right )+e^x (12+8 x) \log (x)+e^{e^{-x} x} \left (-e^x+x^2-x^3+\left (e^x+x-x^2\right ) \log (x)\right )+\left (4 e^x-4 e^x \log (x)\right ) \log \left (2 x^2\right )}{4 e^x x^2+8 e^x x \log (x)+4 e^x \log ^2(x)} \, dx=4\,x+\frac {\frac {x\,\left (3\,x-\ln \left (2\,x^2\right )+2\,\ln \left (x\right )-3\,x^2+5\right )}{x+1}-\frac {x\,\ln \left (x\right )\,\left (6\,x-\ln \left (2\,x^2\right )+2\,\ln \left (x\right )+5\right )}{x+1}}{x+\ln \left (x\right )}+\frac {\ln \left (2\,x^2\right )-2\,\ln \left (x\right )+1}{x+1}+\frac {x\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}}{4\,\left (x+\ln \left (x\right )\right )} \]
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