Integrand size = 170, antiderivative size = 27 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\frac {\left (e^x-x\right ) x}{\log \left (\frac {2}{\left (x+e^x x^2 \log (x)\right )^2}\right )} \]
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\[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 x+2 e^{2 x} x-2 e^x \left (-1+x^2\right )+2 e^x \left (e^x-x\right ) x (2+x) \log (x)+\left (-2 x+e^x (1+x)\right ) \left (1+e^x x \log (x)\right ) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx \\ & = \int \left (\frac {2 \left (1+\log (x)+x \log (x)+x^2 \log (x)+x^2 \log ^2(x)+x^3 \log ^2(x)\right )}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {e^x \left (2+4 \log (x)+2 x \log (x)+\log (x) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )+x \log (x) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )\right )}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}-\frac {2 \left (1+\log (x)+x \log (x)+x^2 \log (x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)+x^2 \log ^2(x) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )\right )}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}\right ) \, dx \\ & = 2 \int \frac {1+\log (x)+x \log (x)+x^2 \log (x)+x^2 \log ^2(x)+x^3 \log ^2(x)}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {1+\log (x)+x \log (x)+x^2 \log (x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)+x^2 \log ^2(x) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \frac {e^x \left (2+4 \log (x)+2 x \log (x)+\log (x) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )+x \log (x) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )\right )}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx \\ & = 2 \int \frac {1+\left (1+x+x^2\right ) \log (x)+x^2 (1+x) \log ^2(x)}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {1+\left (1+x+x^2\right ) \log (x)+x^2 \log ^2(x) \left (2+x+\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )\right )}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \frac {e^x \left (2+\log (x) \left (2 (2+x)+(1+x) \log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )\right )\right )}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx \\ & = 2 \int \left (\frac {x}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {x^2}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {1}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {1}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {1}{x \log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {x}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}\right ) \, dx-2 \int \left (\frac {1+\log (x)+x \log (x)+x^2 \log (x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}\right ) \, dx+\int \left (\frac {2 e^x (1+2 \log (x)+x \log (x))}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {e^x (1+x)}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}\right ) \, dx \\ & = 2 \int \frac {e^x (1+2 \log (x)+x \log (x))}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x^2}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {1+\log (x)+x \log (x)+x^2 \log (x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \frac {e^x (1+x)}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx \\ & = 2 \int \left (\frac {2 e^x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {e^x x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {e^x}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}\right ) \, dx+2 \int \frac {x}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x^2}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {1+\left (1+x+x^2\right ) \log (x)+x^2 (2+x) \log ^2(x)}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \left (\frac {e^x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {e^x x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}\right ) \, dx \\ & = -\left (2 \int \left (\frac {2 x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {x^2}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {1}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {1}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {1}{x \log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}+\frac {x}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )}\right ) \, dx\right )+2 \int \frac {e^x x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {e^x}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x^2}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+4 \int \frac {e^x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \frac {e^x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \frac {e^x x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx \\ & = 2 \int \frac {e^x x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {x^2}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {1}{x \log ^2(x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {1}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {e^x}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {1}{x \log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {x}{\log (x) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x^2}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log ^2(x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {1}{x \log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+2 \int \frac {x}{\log (x) \left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-2 \int \frac {x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+4 \int \frac {e^x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \frac {e^x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx+\int \frac {e^x x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \, dx \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\frac {\left (e^x-x\right ) x}{\log \left (\frac {2}{x^2 \left (1+e^x x \log (x)\right )^2}\right )} \]
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Time = 2.92 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67
method | result | size |
parallelrisch | \(-\frac {-2 \,{\mathrm e}^{x} x +2 x^{2}}{2 \ln \left (\frac {2}{x^{2} \left ({\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{2}+2 x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )}\right )}\) | \(45\) |
risch | \(\frac {2 i \left (x -{\mathrm e}^{x}\right ) x}{\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{\left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )^{2}+\pi {\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )\right ) {\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}\right )}^{2}+\pi {\operatorname {csgn}\left (i \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}\right )}^{3}-\pi \operatorname {csgn}\left (\frac {i}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )^{2}}\right )^{3}-2 i \ln \left (2\right )+4 i \ln \left (x \right )+4 i \ln \left (x \,{\mathrm e}^{x} \ln \left (x \right )+1\right )}\) | \(285\) |
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Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=-\frac {x^{2} - x e^{x}}{\log \left (\frac {2}{x^{4} e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 2 \, x^{3} e^{x} \log \left (x\right ) + x^{2}}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\frac {- x^{2} + x e^{x}}{\log {\left (\frac {2}{x^{4} e^{2 x} \log {\left (x \right )}^{2} + 2 x^{3} e^{x} \log {\left (x \right )} + x^{2}} \right )}} \]
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Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=-\frac {x^{2} - x e^{x}}{\log \left (2\right ) - 2 \, \log \left (x e^{x} \log \left (x\right ) + 1\right ) - 2 \, \log \left (x\right )} \]
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Time = 1.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=-\frac {x^{2} - x e^{x}}{\log \left (2\right ) - \log \left (x^{2} e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 2 \, x e^{x} \log \left (x\right ) + 1\right ) - 2 \, \log \left (x\right )} \]
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Timed out. \[ \int \frac {-2 x+2 e^{2 x} x+e^x \left (2-2 x^2\right )+\left (e^{2 x} \left (4 x+2 x^2\right )+e^x \left (-4 x^2-2 x^3\right )\right ) \log (x)+\left (-2 x+e^x (1+x)+\left (-2 e^x x^2+e^{2 x} \left (x+x^2\right )\right ) \log (x)\right ) \log \left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )}{\left (1+e^x x \log (x)\right ) \log ^2\left (\frac {2}{x^2+2 e^x x^3 \log (x)+e^{2 x} x^4 \log ^2(x)}\right )} \, dx=\int -\frac {2\,x-2\,x\,{\mathrm {e}}^{2\,x}-\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (2\,x^2+4\,x\right )-{\mathrm {e}}^x\,\left (2\,x^3+4\,x^2\right )\right )+\ln \left (\frac {2}{x^2+x^4\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2+2\,x^3\,{\mathrm {e}}^x\,\ln \left (x\right )}\right )\,\left (2\,x+\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,\left (x^2+x\right )\right )-{\mathrm {e}}^x\,\left (x+1\right )\right )+{\mathrm {e}}^x\,\left (2\,x^2-2\right )}{{\ln \left (\frac {2}{x^2+x^4\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2+2\,x^3\,{\mathrm {e}}^x\,\ln \left (x\right )}\right )}^2\,\left (x\,{\mathrm {e}}^x\,\ln \left (x\right )+1\right )} \,d x \]
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