Integrand size = 175, antiderivative size = 33 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log ^2\left (\frac {\log (4 x)}{-2 x^2+\log \left (2-x^2 \left (\frac {e^x}{x}+x\right )\right )}\right ) \]
[Out]
\[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int 2 \left (\frac {1}{x \log (4 x)}+\frac {e^x \left (1+x-4 x^2\right )+x \left (8+3 x-4 x^3\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right ) \, dx \\ & = 2 \int \left (\frac {1}{x \log (4 x)}+\frac {e^x \left (1+x-4 x^2\right )+x \left (8+3 x-4 x^3\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right ) \, dx \\ & = 2 \int \left (-\frac {\left (-2-2 x-2 x^3+x^4\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}-\frac {\left (-2 x^2-\log (4 x)-x \log (4 x)+4 x^2 \log (4 x)+\log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \, dx \\ & = -\left (2 \int \frac {\left (-2-2 x-2 x^3+x^4\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx\right )-2 \int \frac {\left (-2 x^2-\log (4 x)-x \log (4 x)+4 x^2 \log (4 x)+\log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx \\ & = -\left (2 \int \left (-\frac {2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}-\frac {2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}-\frac {2 x^2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}+\frac {x^3 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \, dx\right )-2 \int \left (-\frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )}-\frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}+\frac {4 x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )}-\frac {2 x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}+\frac {\log \left (2-e^x x-x^3\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )}\right ) \, dx \\ & = 2 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )} \, dx+2 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx-2 \int \frac {x^3 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx-2 \int \frac {\log \left (2-e^x x-x^3\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {\log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{x \left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {x^2 \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (-2+e^x x+x^3\right ) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx+4 \int \frac {x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\log (4 x) \left (2 x^2-\log \left (2-e^x x-x^3\right )\right )} \, dx-8 \int \frac {x \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{2 x^2-\log \left (2-e^x x-x^3\right )} \, dx \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log ^2\left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 727, normalized size of antiderivative = 22.03
\[\ln \left (x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}\right )^{2}-2 \ln \left (\ln \left (4 x \right )\right ) \ln \left (x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}\right )+\ln \left (\ln \left (4 x \right )\right )^{2}-2 i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right )-i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )+i \pi \ln \left (\ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}-i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+i \pi \ln \left (\ln \left (4 x \right )\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+2 i \pi \ln \left (\ln \left (4 x \right )\right )-i \pi \ln \left (\ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right ) \operatorname {csgn}\left (i \ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )-i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{3}-2 i \pi \ln \left (\ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+i \pi \ln \left (\ln \left (4 x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{3}+2 i \pi \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right ) \operatorname {csgn}\left (\frac {i \ln \left (4 x \right )}{x^{2}-\frac {\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )}{2}}\right )^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (-{\mathrm e}^{x} x -x^{3}+2\right )-2 x^{2}\right )-2 \ln \left (2\right ) \ln \left (\ln \left (4 x \right )\right )\]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log \left (-\frac {\log \left (4 \, x\right )}{2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )}\right )^{2} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (35) = 70\).
Time = 0.38 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.94 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=-\log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right )^{2} - 2 \, {\left (\log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) - \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )\right )} \log \left (-\frac {\log \left (4 \, x\right )}{2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )}\right ) + 2 \, \log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (35) = 70\).
Time = 0.59 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.61 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log \left (2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )\right )^{2} - 2 \, \log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (-\log \left (4 \, x\right )\right ) + \log \left (-\log \left (4 \, x\right )\right )^{2} - 2 \, \log \left (2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (\log \left (4 \, x\right )\right ) + 2 \, \log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (\log \left (4 \, x\right )\right ) \]
[In]
[Out]
Time = 14.87 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx={\ln \left (\frac {\ln \left (4\,x\right )}{\ln \left (2-x^3-x\,{\mathrm {e}}^x\right )-2\,x^2}\right )}^2 \]
[In]
[Out]