Integrand size = 161, antiderivative size = 31 \[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=e^{-x+\frac {\log \left (\frac {6}{x (-e+x)}\right )}{\log (\log (1-x))}} x \]
[Out]
\[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=\int \frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \left (-\left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )\right )-\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))-\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e-(1+e) x+x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx \\ & = \int \left (\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right )-\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) x+\frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) x \log \left (-\frac {6}{(e-x) x}\right )}{(1-x) \log (1-x) \log ^2(\log (1-x))}+\frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) (-e+2 x)}{(e-x) \log (\log (1-x))}\right ) \, dx \\ & = \int \exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \, dx-\int \exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) x \, dx+\int \frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) x \log \left (-\frac {6}{(e-x) x}\right )}{(1-x) \log (1-x) \log ^2(\log (1-x))} \, dx+\int \frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) (-e+2 x)}{(e-x) \log (\log (1-x))} \, dx \\ & = \int \exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \, dx-\int \exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) x \, dx+\int \left (-\frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \log \left (-\frac {6}{(e-x) x}\right )}{\log (1-x) \log ^2(\log (1-x))}+\frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \log \left (-\frac {6}{(e-x) x}\right )}{(1-x) \log (1-x) \log ^2(\log (1-x))}\right ) \, dx+\int \left (-\frac {2 \exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right )}{\log (\log (1-x))}+\frac {\exp \left (1+\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right )}{(e-x) \log (\log (1-x))}\right ) \, dx \\ & = -\left (2 \int \frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right )}{\log (\log (1-x))} \, dx\right )+\int \exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \, dx-\int \exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) x \, dx-\int \frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \log \left (-\frac {6}{(e-x) x}\right )}{\log (1-x) \log ^2(\log (1-x))} \, dx+\int \frac {\exp \left (\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right ) \log \left (-\frac {6}{(e-x) x}\right )}{(1-x) \log (1-x) \log ^2(\log (1-x))} \, dx+\int \frac {\exp \left (1+\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}\right )}{(e-x) \log (\log (1-x))} \, dx \\ \end{align*}
\[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=\int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 6.19
\[x \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}-x}\right ) \operatorname {csgn}\left (\frac {i}{x \left ({\mathrm e}-x \right )}\right )^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}-x}\right ) \operatorname {csgn}\left (\frac {i}{x \left ({\mathrm e}-x \right )}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi \operatorname {csgn}\left (\frac {i}{x \left ({\mathrm e}-x \right )}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i}{x \left ({\mathrm e}-x \right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-2 i \operatorname {csgn}\left (\frac {i}{x \left ({\mathrm e}-x \right )}\right )^{2} \pi +2 i \pi -2 x \ln \left (\ln \left (1-x \right )\right )-2 \ln \left (x \right )+2 \ln \left (2\right )+2 \ln \left (3\right )-2 \ln \left ({\mathrm e}-x \right )}{2 \ln \left (\ln \left (1-x \right )\right )}}\]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=x e^{\left (-\frac {x \log \left (\log \left (-x + 1\right )\right ) - \log \left (\frac {6}{x^{2} - x e}\right )}{\log \left (\log \left (-x + 1\right )\right )}\right )} \]
[In]
[Out]
Time = 51.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=x e^{\frac {- x \log {\left (\log {\left (1 - x \right )} \right )} + \log {\left (- \frac {6}{- x^{2} + e x} \right )}}{\log {\left (\log {\left (1 - x \right )} \right )}}} \]
[In]
[Out]
\[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=\int { -\frac {{\left ({\left (x^{3} - 2 \, x^{2} - {\left (x^{2} - 2 \, x + 1\right )} e + x\right )} \log \left (-x + 1\right ) \log \left (\log \left (-x + 1\right )\right )^{2} + {\left (2 \, x^{2} - {\left (x - 1\right )} e - 2 \, x\right )} \log \left (-x + 1\right ) \log \left (\log \left (-x + 1\right )\right ) + {\left (x^{2} - x e\right )} \log \left (\frac {6}{x^{2} - x e}\right )\right )} e^{\left (-\frac {x \log \left (\log \left (-x + 1\right )\right ) - \log \left (\frac {6}{x^{2} - x e}\right )}{\log \left (\log \left (-x + 1\right )\right )}\right )}}{{\left (x^{2} - {\left (x - 1\right )} e - x\right )} \log \left (-x + 1\right ) \log \left (\log \left (-x + 1\right )\right )^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=\int { -\frac {{\left ({\left (x^{3} - 2 \, x^{2} - {\left (x^{2} - 2 \, x + 1\right )} e + x\right )} \log \left (-x + 1\right ) \log \left (\log \left (-x + 1\right )\right )^{2} + {\left (2 \, x^{2} - {\left (x - 1\right )} e - 2 \, x\right )} \log \left (-x + 1\right ) \log \left (\log \left (-x + 1\right )\right ) + {\left (x^{2} - x e\right )} \log \left (\frac {6}{x^{2} - x e}\right )\right )} e^{\left (-\frac {x \log \left (\log \left (-x + 1\right )\right ) - \log \left (\frac {6}{x^{2} - x e}\right )}{\log \left (\log \left (-x + 1\right )\right )}\right )}}{{\left (x^{2} - {\left (x - 1\right )} e - x\right )} \log \left (-x + 1\right ) \log \left (\log \left (-x + 1\right )\right )^{2}} \,d x } \]
[In]
[Out]
Time = 8.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {\log \left (-\frac {6}{e x-x^2}\right )-x \log (\log (1-x))}{\log (\log (1-x))}} \left (\left (-e x+x^2\right ) \log \left (-\frac {6}{e x-x^2}\right )+\left (e (1-x)-2 x+2 x^2\right ) \log (1-x) \log (\log (1-x))+\left (x-2 x^2+x^3+e \left (-1+2 x-x^2\right )\right ) \log (1-x) \log ^2(\log (1-x))\right )}{\left (e (-1+x)+x-x^2\right ) \log (1-x) \log ^2(\log (1-x))} \, dx=x\,{\mathrm {e}}^{-x}\,{\left (-\frac {6}{x\,\mathrm {e}-x^2}\right )}^{\frac {1}{\ln \left (\ln \left (1-x\right )\right )}} \]
[In]
[Out]