Integrand size = 142, antiderivative size = 23 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=13-2 x+\frac {1}{\log ^2\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \]
[Out]
\[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=\int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-4 \left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right )+2 x \log (x) \left (4 x+\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )\right )}{x \left (2-x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ & = \int \left (-2-\frac {4 \left (2-x^2+2 x^2 \log (x)+2 \log \left (-2+x^2\right )-x^2 \log \left (-2+x^2\right )\right )}{x \left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx \\ & = -2 x-4 \int \frac {2-x^2+2 x^2 \log (x)+2 \log \left (-2+x^2\right )-x^2 \log \left (-2+x^2\right )}{x \left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ & = -2 x-4 \int \left (\frac {-2+x^2-2 x^2 \log (x)-2 \log \left (-2+x^2\right )+x^2 \log \left (-2+x^2\right )}{2 x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x-x^3+2 x^3 \log (x)+2 x \log \left (-2+x^2\right )-x^3 \log \left (-2+x^2\right )}{2 \left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx \\ & = -2 x-2 \int \frac {-2+x^2-2 x^2 \log (x)-2 \log \left (-2+x^2\right )+x^2 \log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-2 \int \frac {2 x-x^3+2 x^3 \log (x)+2 x \log \left (-2+x^2\right )-x^3 \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ & = -2 x-2 \int \left (\frac {2 x^3}{\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {x^3}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {x^3 \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx-2 \int \frac {-2 x^2 \log (x)+\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ & = -2 x-2 \int \left (-\frac {2 x}{\left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {2}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {x}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {2 \log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {x \log \left (-2+x^2\right )}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx+2 \int \frac {x^3}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+2 \int \frac {x^3 \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \frac {x^3}{\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \frac {x}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \frac {x \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ & = -2 x+2 \int \left (\frac {x}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx+2 \int \left (\frac {x \log \left (-2+x^2\right )}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx-2 \int \frac {x}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-2 \int \frac {x \log \left (-2+x^2\right )}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \left (\frac {x}{\left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x}{\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx-4 \int \left (-\frac {1}{2 \left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {1}{2 \left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx-4 \int \left (-\frac {\log \left (-2+x^2\right )}{2 \left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {\log \left (-2+x^2\right )}{2 \left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx+4 \int \frac {x}{\left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {1}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {\log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ & = -2 x+2 \int \frac {1}{\left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-2 \int \frac {1}{\left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+2 \int \frac {\log \left (-2+x^2\right )}{\left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-2 \int \frac {\log \left (-2+x^2\right )}{\left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {1}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {x}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {\log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {x \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-8 \int \frac {x}{\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ & = -2 x+2 \int \frac {1}{\left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-2 \int \frac {1}{\left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+2 \int \frac {\log \left (-2+x^2\right )}{\left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-2 \int \frac {\log \left (-2+x^2\right )}{\left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \left (-\frac {1}{2 \left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {1}{2 \left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx+4 \int \left (-\frac {\log \left (-2+x^2\right )}{2 \left (\sqrt {2}-x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {\log \left (-2+x^2\right )}{2 \left (\sqrt {2}+x\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx+4 \int \frac {1}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {\log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-8 \int \left (-\frac {1}{2 \left (\sqrt {2}-x\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {1}{2 \left (\sqrt {2}+x\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx \\ & = -2 x+4 \int \frac {1}{\left (\sqrt {2}-x\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \frac {1}{\left (\sqrt {2}+x\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {1}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+4 \int \frac {\log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-2 x+\frac {1}{\log ^2\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(23)=46\).
Time = 39.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.61
method | result | size |
parallelrisch | \(\frac {4-8 {\ln \left (\frac {\ln \left (x^{2}-2\right )^{2}+2 \ln \left (x^{2}-2\right )+1}{\ln \left (x \right )^{2}}\right )}^{2} x}{4 {\ln \left (\frac {\ln \left (x^{2}-2\right )^{2}+2 \ln \left (x^{2}-2\right )+1}{\ln \left (x \right )^{2}}\right )}^{2}}\) | \(60\) |
risch | \(-2 x -\frac {4}{{\left (\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right ) \operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )}^{2}-\pi \operatorname {csgn}\left (i \ln \left (x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}-\pi \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}+\pi {\operatorname {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )\right )}^{2} \operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )\right ) {\operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right )}^{2}+\pi {\operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right )}^{3}-\pi \,\operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right ) {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )}^{3}-4 i \ln \left (\ln \left (x \right )\right )+4 i \ln \left (\ln \left (x^{2}-2\right )+1\right )\right )}^{2}}\) | \(290\) |
default | \(\text {Expression too large to display}\) | \(13907\) |
parts | \(\text {Expression too large to display}\) | \(13907\) |
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-\frac {2 \, x \log \left (\frac {\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1}{\log \left (x\right )^{2}}\right )^{2} - 1}{\log \left (\frac {\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1}{\log \left (x\right )^{2}}\right )^{2}} \]
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Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=- 2 x + \frac {1}{\log {\left (\frac {\log {\left (x^{2} - 2 \right )}^{2} + 2 \log {\left (x^{2} - 2 \right )} + 1}{\log {\left (x \right )}^{2}} \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-\frac {8 \, x \log \left (\log \left (x^{2} - 2\right ) + 1\right )^{2} - 16 \, x \log \left (\log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )\right ) + 8 \, x \log \left (\log \left (x\right )\right )^{2} - 1}{4 \, {\left (\log \left (\log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, \log \left (\log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (23) = 46\).
Time = 44.41 (sec) , antiderivative size = 393, normalized size of antiderivative = 17.09 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-2 \, x + \frac {x^{2} \log \left (x^{2} - 2\right ) - 2 \, x^{2} \log \left (x\right ) + x^{2} - 2 \, \log \left (x^{2} - 2\right ) - 2}{x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) + 4 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} \log \left (x\right ) + x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) + x^{2} \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} + 4 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) - 2 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} + 4 \, \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) - 2 \, \log \left (\log \left (x\right )^{2}\right )^{2}} \]
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Time = 12.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=\frac {1}{{\ln \left (\frac {{\ln \left (x^2-2\right )}^2+2\,\ln \left (x^2-2\right )+1}{{\ln \left (x\right )}^2}\right )}^2}-2\,x \]
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