Integrand size = 245, antiderivative size = 26 \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=x \log \left (\frac {\log \left (x-\frac {x^3}{\left (-e^x+x\right )^2}\right )}{\log (\log (x))}\right ) \]
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Time = 2.74 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 17, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6820, 6874, 2629} \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=x \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right ) \]
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Rule 2629
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{2 x}-3 e^x x+2 x^3}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}-\frac {1}{\log (x) \log (\log (x))}+\log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right )\right ) \, dx \\ & = \int \frac {e^{2 x}-3 e^x x+2 x^3}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx-\int \frac {1}{\log (x) \log (\log (x))} \, dx+\int \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right ) \, dx \\ & = x \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right )+\int \left (\frac {1}{\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}+\frac {2 (-1+x) x}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}-\frac {2 (-1+x) x}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}\right ) \, dx-\int \frac {\frac {e^{2 x}-3 e^x x+2 x^3}{\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}-\frac {e^{2 x}-3 e^x x+2 x^2}{\log (x) \log (\log (x))}}{\left (-e^x+x\right ) \left (-e^x+2 x\right )} \, dx-\int \frac {1}{\log (x) \log (\log (x))} \, dx \\ & = x \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right )+2 \int \frac {(-1+x) x}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx-2 \int \frac {(-1+x) x}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx+\int \frac {1}{\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx-\int \frac {1}{\log (x) \log (\log (x))} \, dx-\int \left (\frac {2 (-1+x) x}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}-\frac {2 (-1+x) x}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}+\frac {-\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )+\log (x) \log (\log (x))}{\log (x) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right ) \log (\log (x))}\right ) \, dx \\ & = x \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right )+2 \int \left (-\frac {x}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}+\frac {x^2}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}\right ) \, dx-2 \int \left (-\frac {x}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}+\frac {x^2}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}\right ) \, dx-2 \int \frac {(-1+x) x}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx+2 \int \frac {(-1+x) x}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx+\int \frac {1}{\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx-\int \frac {1}{\log (x) \log (\log (x))} \, dx-\int \frac {-\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )+\log (x) \log (\log (x))}{\log (x) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right ) \log (\log (x))} \, dx \\ & = x \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right )-2 \int \left (-\frac {x}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}+\frac {x^2}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}\right ) \, dx+2 \int \left (-\frac {x}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}+\frac {x^2}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}\right ) \, dx-2 \int \frac {x}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx+2 \int \frac {x}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx+2 \int \frac {x^2}{\left (e^x-2 x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx-2 \int \frac {x^2}{\left (e^x-x\right ) \log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx+\int \frac {1}{\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )} \, dx-\int \left (\frac {1}{\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}-\frac {1}{\log (x) \log (\log (x))}\right ) \, dx-\int \frac {1}{\log (x) \log (\log (x))} \, dx \\ & = x \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right ) \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=x \log \left (\frac {\log \left (\frac {e^x \left (e^x-2 x\right ) x}{\left (e^x-x\right )^2}\right )}{\log (\log (x))}\right ) \]
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\[\int \frac {\left ({\mathrm e}^{2 x}-3 \,{\mathrm e}^{x} x +2 x^{2}\right ) \ln \left (x \right ) \ln \left (\frac {x \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x^{2}}{{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x +x^{2}}\right ) \ln \left (\ln \left (x \right )\right ) \ln \left (\frac {\ln \left (\frac {x \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x^{2}}{{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x +x^{2}}\right )}{\ln \left (\ln \left (x \right )\right )}\right )+\left ({\mathrm e}^{2 x}-3 \,{\mathrm e}^{x} x +2 x^{3}\right ) \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+\left (-{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x} x -2 x^{2}\right ) \ln \left (\frac {x \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x^{2}}{{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x +x^{2}}\right )}{\left ({\mathrm e}^{2 x}-3 \,{\mathrm e}^{x} x +2 x^{2}\right ) \ln \left (x \right ) \ln \left (\frac {x \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x^{2}}{{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x +x^{2}}\right ) \ln \left (\ln \left (x \right )\right )}d x\]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=x \log \left (\frac {\log \left (-\frac {2 \, x^{2} e^{x} - x e^{\left (2 \, x\right )}}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}}\right )}{\log \left (\log \left (x\right )\right )}\right ) \]
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Timed out. \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=\text {Timed out} \]
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Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=x \log \left (x + \log \left (x\right ) - 2 \, \log \left (-x + e^{x}\right ) + \log \left (-2 \, x + e^{x}\right )\right ) - x \log \left (\log \left (\log \left (x\right )\right )\right ) \]
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Exception generated. \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=\text {Exception raised: TypeError} \]
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Time = 13.62 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {\left (-e^{2 x}+3 e^x x-2 x^2\right ) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )+\left (e^{2 x}-3 e^x x+2 x^3\right ) \log (x) \log (\log (x))+\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x)) \log \left (\frac {\log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right )}{\log (\log (x))}\right )}{\left (e^{2 x}-3 e^x x+2 x^2\right ) \log (x) \log \left (\frac {e^{2 x} x-2 e^x x^2}{e^{2 x}-2 e^x x+x^2}\right ) \log (\log (x))} \, dx=x\,\ln \left (\frac {\ln \left (\frac {x\,{\mathrm {e}}^{2\,x}-2\,x^2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2}\right )}{\ln \left (\ln \left (x\right )\right )}\right ) \]
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