Integrand size = 364, antiderivative size = 23 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log \left (\frac {x}{e^{\left (x+\frac {x}{x+\log (\log (x))}\right )^2}+\log (x)}\right ) \]
[Out]
\[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+\exp \left (\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}\right ) \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+\exp \left (\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}\right ) \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+3 x^2 (-1+\log (x)) \log (x) \log (\log (x))+3 x (-1+\log (x)) \log (x) \log ^2(\log (x))+(-1+\log (x)) \log (x) \log ^3(\log (x))+\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right ) \left (2 x^2 (1+x+\log (\log (x)))+\log (x) \left (x^3-2 x^4-2 x^5+\left (x^2-6 x^3-6 x^4\right ) \log (\log (x))+x \left (3-4 x-6 x^2\right ) \log ^2(\log (x))+\left (1-2 x^2\right ) \log ^3(\log (x))\right )\right )}{x \log (x) \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx \\ & = \int \left (\frac {2 x^2+2 x^3+x^3 \log (x)-2 x^4 \log (x)-2 x^5 \log (x)+2 x^2 \log (\log (x))+x^2 \log (x) \log (\log (x))-6 x^3 \log (x) \log (\log (x))-6 x^4 \log (x) \log (\log (x))+3 x \log (x) \log ^2(\log (x))-4 x^2 \log (x) \log ^2(\log (x))-6 x^3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x))-2 x^2 \log (x) \log ^3(\log (x))}{x \log (x) (x+\log (\log (x)))^3}+\frac {-2 x^2-3 x^3+2 x^4 \log (x)+2 x^5 \log (x)-5 x^2 \log (\log (x))+2 x^2 \log (x) \log (\log (x))+6 x^3 \log (x) \log (\log (x))+6 x^4 \log (x) \log (\log (x))-3 x \log ^2(\log (x))+4 x^2 \log (x) \log ^2(\log (x))+6 x^3 \log (x) \log ^2(\log (x))-\log ^3(\log (x))+2 x^2 \log (x) \log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}\right ) \, dx \\ & = \int \frac {2 x^2+2 x^3+x^3 \log (x)-2 x^4 \log (x)-2 x^5 \log (x)+2 x^2 \log (\log (x))+x^2 \log (x) \log (\log (x))-6 x^3 \log (x) \log (\log (x))-6 x^4 \log (x) \log (\log (x))+3 x \log (x) \log ^2(\log (x))-4 x^2 \log (x) \log ^2(\log (x))-6 x^3 \log (x) \log ^2(\log (x))+\log (x) \log ^3(\log (x))-2 x^2 \log (x) \log ^3(\log (x))}{x \log (x) (x+\log (\log (x)))^3} \, dx+\int \frac {-2 x^2-3 x^3+2 x^4 \log (x)+2 x^5 \log (x)-5 x^2 \log (\log (x))+2 x^2 \log (x) \log (\log (x))+6 x^3 \log (x) \log (\log (x))+6 x^4 \log (x) \log (\log (x))-3 x \log ^2(\log (x))+4 x^2 \log (x) \log ^2(\log (x))+6 x^3 \log (x) \log ^2(\log (x))-\log ^3(\log (x))+2 x^2 \log (x) \log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx \\ & = \int \frac {-x^2 (2+3 x)-5 x^2 \log (\log (x))-3 x \log ^2(\log (x))-\log ^3(\log (x))+2 x^2 \log (x) \left (x^2 (1+x)+\left (1+3 x+3 x^2\right ) \log (\log (x))+(2+3 x) \log ^2(\log (x))+\log ^3(\log (x))\right )}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+\int \frac {2 x^2 (1+x+\log (\log (x)))+\log (x) \left (x^3-2 x^4-2 x^5+\left (x^2-6 x^3-6 x^4\right ) \log (\log (x))+x \left (3-4 x-6 x^2\right ) \log ^2(\log (x))+\left (1-2 x^2\right ) \log ^3(\log (x))\right )}{x \log (x) (x+\log (\log (x)))^3} \, dx \\ & = \int \left (-\frac {2 x}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}-\frac {3 x^2}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {2 x^3 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {2 x^4 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}-\frac {5 x \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {2 x \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {6 x^2 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {6 x^3 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}-\frac {3 \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {4 x \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {6 x^2 \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}-\frac {\log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}+\frac {2 x \log (x) \log ^3(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3}\right ) \, dx+\int \left (\frac {1-2 x^2}{x}+\frac {2 x (1+x \log (x))}{\log (x) (x+\log (\log (x)))^3}+\frac {2 \left (x-x \log (x)+x^2 \log (x)\right )}{\log (x) (x+\log (\log (x)))^2}-\frac {4 x}{x+\log (\log (x))}\right ) \, dx \\ & = -\left (2 \int \frac {x}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx\right )+2 \int \frac {x^3 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x^4 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x (1+x \log (x))}{\log (x) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log ^3(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x-x \log (x)+x^2 \log (x)}{\log (x) (x+\log (\log (x)))^2} \, dx-3 \int \frac {x^2}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-3 \int \frac {\log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+4 \int \frac {x \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-4 \int \frac {x}{x+\log (\log (x))} \, dx-5 \int \frac {x \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^3 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+\int \frac {1-2 x^2}{x} \, dx-\int \frac {\log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx \\ & = -\left (2 \int \frac {x}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx\right )+2 \int \frac {x^3 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x^4 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log ^3(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x+(-1+x) x \log (x)}{\log (x) (x+\log (\log (x)))^2} \, dx+2 \int \left (\frac {x^2}{(x+\log (\log (x)))^3}+\frac {x}{\log (x) (x+\log (\log (x)))^3}\right ) \, dx-3 \int \frac {x^2}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-3 \int \frac {\log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+4 \int \frac {x \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-4 \int \frac {x}{x+\log (\log (x))} \, dx-5 \int \frac {x \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^3 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+\int \left (\frac {1}{x}-2 x\right ) \, dx-\int \frac {\log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx \\ & = -x^2+\log (x)+2 \int \frac {x^2}{(x+\log (\log (x)))^3} \, dx+2 \int \frac {x}{\log (x) (x+\log (\log (x)))^3} \, dx-2 \int \frac {x}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x^3 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x^4 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log ^3(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \left (-\frac {x}{(x+\log (\log (x)))^2}+\frac {x^2}{(x+\log (\log (x)))^2}+\frac {x}{\log (x) (x+\log (\log (x)))^2}\right ) \, dx-3 \int \frac {x^2}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-3 \int \frac {\log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+4 \int \frac {x \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-4 \int \frac {x}{x+\log (\log (x))} \, dx-5 \int \frac {x \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^3 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-\int \frac {\log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx \\ & = -x^2+\log (x)+2 \int \frac {x^2}{(x+\log (\log (x)))^3} \, dx+2 \int \frac {x}{\log (x) (x+\log (\log (x)))^3} \, dx-2 \int \frac {x}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x^3 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x^4 \log (x)}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+2 \int \frac {x \log (x) \log ^3(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-2 \int \frac {x}{(x+\log (\log (x)))^2} \, dx+2 \int \frac {x^2}{(x+\log (\log (x)))^2} \, dx+2 \int \frac {x}{\log (x) (x+\log (\log (x)))^2} \, dx-3 \int \frac {x^2}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-3 \int \frac {\log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+4 \int \frac {x \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-4 \int \frac {x}{x+\log (\log (x))} \, dx-5 \int \frac {x \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^3 \log (x) \log (\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx+6 \int \frac {x^2 \log (x) \log ^2(\log (x))}{\left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx-\int \frac {\log ^3(\log (x))}{x \left (\exp \left (\frac {x^2 (1+x+\log (\log (x)))^2}{(x+\log (\log (x)))^2}\right )+\log (x)\right ) (x+\log (\log (x)))^3} \, dx \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log (x)-\log \left (e^{x^2+\frac {x^2}{(x+\log (\log (x)))^2}+\frac {2 x^2}{x+\log (\log (x))}}+\log (x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(22)=44\).
Time = 85.66 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43
method | result | size |
parallelrisch | \(\ln \left (x \right )-\ln \left (\ln \left (x \right )+{\mathrm e}^{\frac {x^{2} \left (\ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (\ln \left (x \right )\right )+x^{2}+2 \ln \left (\ln \left (x \right )\right )+2 x +1\right )}{\ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (\ln \left (x \right )\right )+x^{2}}}\right )\) | \(56\) |
risch | \(\ln \left (x \right )-x^{2}-\frac {\left (2 x +2 \ln \left (\ln \left (x \right )\right )+1\right ) x^{2}}{\left (\ln \left (\ln \left (x \right )\right )+x \right )^{2}}+\frac {x^{2} \ln \left (\ln \left (x \right )\right )^{2}+\left (2 x^{3}+2 x^{2}\right ) \ln \left (\ln \left (x \right )\right )+x^{4}+2 x^{3}+x^{2}}{\ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (\ln \left (x \right )\right )+x^{2}}-\ln \left (\ln \left (x \right )+{\mathrm e}^{\frac {x^{2} \left (\ln \left (\ln \left (x \right )\right )+1+x \right )^{2}}{\left (\ln \left (\ln \left (x \right )\right )+x \right )^{2}}}\right )\) | \(111\) |
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log \left (x\right ) - \log \left (e^{\left (\frac {x^{4} + x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + x^{2}\right )} \log \left (\log \left (x\right )\right )}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}}\right )} + \log \left (x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
Time = 1.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.83 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\log {\left (x \right )} - \log {\left (e^{\frac {x^{4} + 2 x^{3} + x^{2} \log {\left (\log {\left (x \right )} \right )}^{2} + x^{2} + \left (2 x^{3} + 2 x^{2}\right ) \log {\left (\log {\left (x \right )} \right )}}{x^{2} + 2 x \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (x \right )} \right )}^{2}}} + \log {\left (x \right )} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (22) = 44\).
Time = 0.54 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.39 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=-\frac {x^{3} + x^{2} \log \left (\log \left (x\right )\right ) + 2 \, x^{2} + 2 \, x}{x + \log \left (\log \left (x\right )\right )} - \log \left ({\left (\log \left (x\right )^{\frac {2}{x + \log \left (\log \left (x\right )\right )}} \log \left (x\right )^{3} + e^{\left (x^{2} + 2 \, x + \frac {\log \left (\log \left (x\right )\right )^{2}}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}} + \frac {2 \, \log \left (\log \left (x\right )\right )^{2}}{x + \log \left (\log \left (x\right )\right )} + 1\right )}\right )} e^{\left (-x^{2} - 2 \, x - \frac {2 \, \log \left (\log \left (x\right )\right )^{2}}{x + \log \left (\log \left (x\right )\right )} - 1\right )}\right ) + \log \left (x\right ) \]
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\[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\int { \frac {x^{3} \log \left (x\right )^{2} - x^{3} \log \left (x\right ) + {\left (\log \left (x\right )^{2} - \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{3} + 3 \, {\left (x \log \left (x\right )^{2} - x \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{2} - {\left ({\left (2 \, x^{2} - 1\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + {\left (6 \, x^{3} + 4 \, x^{2} - 3 \, x\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, x^{3} - 2 \, x^{2} + {\left (2 \, x^{5} + 2 \, x^{4} - x^{3}\right )} \log \left (x\right ) - {\left (2 \, x^{2} - {\left (6 \, x^{4} + 6 \, x^{3} - x^{2}\right )} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )\right )} e^{\left (\frac {x^{4} + x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + x^{2}\right )} \log \left (\log \left (x\right )\right )}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}}\right )} + 3 \, {\left (x^{2} \log \left (x\right )^{2} - x^{2} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )}{x^{4} \log \left (x\right )^{2} + 3 \, x^{3} \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) + 3 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x\right )\right )^{2} + x \log \left (x\right )^{2} \log \left (\log \left (x\right )\right )^{3} + {\left (x^{4} \log \left (x\right ) + 3 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 3 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + x \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3}\right )} e^{\left (\frac {x^{4} + x^{2} \log \left (\log \left (x\right )\right )^{2} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{3} + x^{2}\right )} \log \left (\log \left (x\right )\right )}{x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}}\right )}} \,d x } \]
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Time = 10.54 (sec) , antiderivative size = 156, normalized size of antiderivative = 6.78 \[ \int \frac {-x^3 \log (x)+x^3 \log ^2(x)+\left (-3 x^2 \log (x)+3 x^2 \log ^2(x)\right ) \log (\log (x))+\left (-3 x \log (x)+3 x \log ^2(x)\right ) \log ^2(\log (x))+\left (-\log (x)+\log ^2(x)\right ) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (2 x^2+2 x^3+\left (x^3-2 x^4-2 x^5\right ) \log (x)+\left (2 x^2+\left (x^2-6 x^3-6 x^4\right ) \log (x)\right ) \log (\log (x))+\left (3 x-4 x^2-6 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-2 x^2\right ) \log (x) \log ^3(\log (x))\right )}{x^4 \log ^2(x)+3 x^3 \log ^2(x) \log (\log (x))+3 x^2 \log ^2(x) \log ^2(\log (x))+x \log ^2(x) \log ^3(\log (x))+e^{\frac {x^2+2 x^3+x^4+\left (2 x^2+2 x^3\right ) \log (\log (x))+x^2 \log ^2(\log (x))}{x^2+2 x \log (\log (x))+\log ^2(\log (x))}} \left (x^4 \log (x)+3 x^3 \log (x) \log (\log (x))+3 x^2 \log (x) \log ^2(\log (x))+x \log (x) \log ^3(\log (x))\right )} \, dx=\ln \left (x\right )-\ln \left (\ln \left (x\right )+{\mathrm {e}}^{\frac {x^2}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {x^4}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^3}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^2\,\ln \left (\ln \left (x\right )\right )}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {2\,x^3\,\ln \left (\ln \left (x\right )\right )}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\,{\mathrm {e}}^{\frac {x^2\,{\ln \left (\ln \left (x\right )\right )}^2}{x^2+2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2}}\right ) \]
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