Integrand size = 124, antiderivative size = 31 \[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\frac {\frac {1}{x}+\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{\log (5) \log \left (2 x^2\right )} \]
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\[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 x \log (x)-\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )-\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))-\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx \\ & = \frac {\int \frac {-2 x \log (x)-\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )-\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))-\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)} \\ & = \frac {\int \left (\frac {-2 x \log (x)+x \log \left (2 x^2\right )-x \log (x) \log \left (2 x^2\right )-x^2 \log (x) \log \left (2 x^2\right )+x^3 \log (x) \log \left (2 x^2\right )+2 \log (x) \log (\log (x))+\log (x) \log \left (2 x^2\right ) \log (\log (x))-x^2 \log (x) \log \left (2 x^2\right ) \log (\log (x))}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))}-\frac {2 \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )}\right ) \, dx}{\log (5)} \\ & = \frac {\int \frac {-2 x \log (x)+x \log \left (2 x^2\right )-x \log (x) \log \left (2 x^2\right )-x^2 \log (x) \log \left (2 x^2\right )+x^3 \log (x) \log \left (2 x^2\right )+2 \log (x) \log (\log (x))+\log (x) \log \left (2 x^2\right ) \log (\log (x))-x^2 \log (x) \log \left (2 x^2\right ) \log (\log (x))}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {\int \frac {x \log \left (2 x^2\right )+\log (x) \left (-2 x+2 \log (\log (x))+\log \left (2 x^2\right ) \left (x \left (-1-x+x^2\right )-\left (-1+x^2\right ) \log (\log (x))\right )\right )}{x^2 \log (x) \log ^2\left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {\int \left (\frac {-2-\log \left (2 x^2\right )+x^2 \log \left (2 x^2\right )}{x^2 \log ^2\left (2 x^2\right )}+\frac {1-x \log (x)}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))}\right ) \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {\int \frac {-2-\log \left (2 x^2\right )+x^2 \log \left (2 x^2\right )}{x^2 \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {\int \frac {1-x \log (x)}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {\int \frac {-2+\left (-1+x^2\right ) \log \left (2 x^2\right )}{x^2 \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {\int \left (-\frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))}+\frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))}\right ) \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {\int \left (-\frac {2}{x^2 \log ^2\left (2 x^2\right )}+\frac {-1+x^2}{x^2 \log \left (2 x^2\right )}\right ) \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {\int \frac {-1+x^2}{x^2 \log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {1}{x^2 \log ^2\left (2 x^2\right )} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {1}{x \log (5) \log \left (2 x^2\right )}+\frac {\int \left (\frac {1}{\log \left (2 x^2\right )}-\frac {1}{x^2 \log \left (2 x^2\right )}\right ) \, dx}{\log (5)}+\frac {\int \frac {1}{x^2 \log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ & = \frac {1}{x \log (5) \log \left (2 x^2\right )}+\frac {\int \frac {1}{\log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{x^2 \log \left (2 x^2\right )} \, dx}{\log (5)}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (2 x^2\right )\right )}{\sqrt {2} x \log (5)} \\ & = \frac {\sqrt {x^2} \operatorname {ExpIntegralEi}\left (-\frac {1}{2} \log \left (2 x^2\right )\right )}{\sqrt {2} x \log (5)}+\frac {1}{x \log (5) \log \left (2 x^2\right )}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)}+\frac {x \text {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (2 x^2\right )\right )}{2 \sqrt {2} \sqrt {x^2} \log (5)}-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (2 x^2\right )\right )}{\sqrt {2} x \log (5)} \\ & = \frac {x \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left (2 x^2\right )\right )}{2 \sqrt {2} \sqrt {x^2} \log (5)}+\frac {1}{x \log (5) \log \left (2 x^2\right )}-\frac {\int \frac {1}{\log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}+\frac {\int \frac {1}{x \log (x) \log \left (2 x^2\right ) (x-\log (\log (x)))} \, dx}{\log (5)}-\frac {2 \int \frac {\log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log ^2\left (2 x^2\right )} \, dx}{\log (5)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\frac {1+x \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{x \log (5) \log \left (2 x^2\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 9.45
\[\frac {2 i \ln \left ({\mathrm e}^{x}\right )}{\ln \left (5\right ) \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \ln \left (2\right )+4 i \ln \left (x \right )\right )}+\frac {-2 i x \ln \left (x -\ln \left (\ln \left (x \right )\right )\right )+2 \pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \left (x \right )\right )}\right )^{2}-\pi x \,\operatorname {csgn}\left (\frac {i}{x -\ln \left (\ln \left (x \right )\right )}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \left (x \right )\right )}\right )^{2}+\pi x \,\operatorname {csgn}\left (\frac {i}{x -\ln \left (\ln \left (x \right )\right )}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \left (x \right )\right )}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-\pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \left (x \right )\right )}\right )^{3}-\pi x \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\ln \left (\ln \left (x \right )\right )}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-2 \pi x +2 i}{\ln \left (5\right ) \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \ln \left (2\right )+4 i \ln \left (x \right )\right ) x}\]
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\frac {x \log \left (-\frac {e^{x}}{x - \log \left (\log \left (x\right )\right )}\right ) + 1}{x \log \left (5\right ) \log \left (2\right ) + 2 \, x \log \left (5\right ) \log \left (x\right )} \]
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Exception generated. \[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\frac {x^{2} - x \log \left (-x + \log \left (\log \left (x\right )\right )\right ) + 1}{x \log \left (5\right ) \log \left (2\right ) + 2 \, x \log \left (5\right ) \log \left (x\right )} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\frac {i \, \pi x + x^{2} + 1}{x \log \left (5\right ) \log \left (2\right ) + 2 \, x \log \left (5\right ) \log \left (x\right )} - \frac {\log \left (x - \log \left (\log \left (x\right )\right )\right )}{\log \left (5\right ) \log \left (2\right ) + 2 \, \log \left (5\right ) \log \left (x\right )} \]
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Time = 14.52 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {2 x \log (x)+\left (-x+\left (x+x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+\left (-2 \log (x)+\left (-1+x^2\right ) \log (x) \log \left (2 x^2\right )\right ) \log (\log (x))+\left (2 x^2 \log (x)-2 x \log (x) \log (\log (x))\right ) \log \left (\frac {e^x}{-x+\log (\log (x))}\right )}{-x^3 \log (5) \log (x) \log ^2\left (2 x^2\right )+x^2 \log (5) \log (x) \log ^2\left (2 x^2\right ) \log (\log (x))} \, dx=\frac {x\,\ln \left (-\frac {{\mathrm {e}}^x}{x-\ln \left (\ln \left (x\right )\right )}\right )+1}{x\,\ln \left (5\right )\,\ln \left (2\,x^2\right )} \]
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