Integrand size = 326, antiderivative size = 28 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=e^4+x^2+\frac {x}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \]
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\[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+\log \left (\frac {1}{x}\right ) \left (1+8 x^3+2 x \log ^2(x)+\left (1+8 x^2\right ) \log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )+2 x \log ^2\left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )-\log (x) \left (1+8 x^2+4 x \log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )\right )+2 \log ^2\left (\frac {1}{x}\right ) \left (1+8 x^3+2 x \log ^2(x)+\left (1+8 x^2\right ) \log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )+2 x \log ^2\left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )-\log (x) \left (1+8 x^2+4 x \log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )\right )}{\log \left (\frac {1}{x}\right ) \left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2} \, dx \\ & = \int \left (2 x+\frac {-1+\log \left (\frac {1}{x}\right )-2 x \log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )-4 x \log ^2\left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right ) \left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}+\frac {1}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )}\right ) \, dx \\ & = x^2+\int \frac {-1+\log \left (\frac {1}{x}\right )-2 x \log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )-4 x \log ^2\left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right ) \left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2} \, dx+\int \frac {1}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \, dx \\ & = x^2+\int \frac {1}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \, dx+\int \left (\frac {1}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}-\frac {2 x}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}-\frac {1}{\log \left (\frac {1}{x}\right ) \left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}+\frac {2 \log \left (\frac {1}{x}\right )}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}-\frac {4 x \log \left (\frac {1}{x}\right )}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2}\right ) \, dx \\ & = x^2-2 \int \frac {x}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2} \, dx+2 \int \frac {\log \left (\frac {1}{x}\right )}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2} \, dx-4 \int \frac {x \log \left (\frac {1}{x}\right )}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2} \, dx+\int \frac {1}{\left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2} \, dx-\int \frac {1}{\log \left (\frac {1}{x}\right ) \left (1+2 \log \left (\frac {1}{x}\right )\right ) \left (2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )\right )^2} \, dx+\int \frac {1}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \, dx \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=x^2+\frac {x}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(27)=54\).
Time = 7.62 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04
method | result | size |
parallelrisch | \(\frac {-48 \ln \left (\frac {2 \ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )+192 x^{3}+48 \ln \left (x \right )-96 x^{2} \ln \left (x \right )+96 \ln \left (\frac {2 \ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right ) x^{2}}{192 x -96 \ln \left (x \right )+96 \ln \left (\frac {2 \ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}\) | \(85\) |
risch | \(x^{2}+\frac {2 x}{-i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )-\frac {1}{2}\right )}{\ln \left (x \right )}\right )^{3}+2 \ln \left (2\right )+4 x -2 \ln \left (x \right )-2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (\ln \left (x \right )-\frac {1}{2}\right )}\) | \(136\) |
derivativedivides | \(x^{2}+\frac {2 i}{\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )}{x}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}-\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}+\frac {\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{3}}{x}+\frac {2 i \ln \left (2\right )}{x}-\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )\right )}{x}+\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{x}+\frac {2 i \ln \left (\frac {1}{x}\right )}{x}+4 i}\) | \(187\) |
default | \(x^{2}+\frac {2 i}{\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )}{x}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\frac {1}{x}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}-\frac {\pi \,\operatorname {csgn}\left (i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{2}}{x}+\frac {\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{\ln \left (\frac {1}{x}\right )}\right )^{3}}{x}+\frac {2 i \ln \left (2\right )}{x}-\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )\right )}{x}+\frac {2 i \ln \left (\ln \left (\frac {1}{x}\right )+\frac {1}{2}\right )}{x}+\frac {2 i \ln \left (\frac {1}{x}\right )}{x}+4 i}\) | \(187\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\frac {2 \, x^{3} + x^{2} \log \left (\frac {2 \, \log \left (\frac {1}{x}\right ) + 1}{\log \left (\frac {1}{x}\right )}\right ) + x^{2} \log \left (\frac {1}{x}\right ) + x}{2 \, x + \log \left (\frac {2 \, \log \left (\frac {1}{x}\right ) + 1}{\log \left (\frac {1}{x}\right )}\right ) + \log \left (\frac {1}{x}\right )} \]
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Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=x^{2} + \frac {x}{2 x - \log {\left (x \right )} + \log {\left (- \frac {1 - 2 \log {\left (x \right )}}{\log {\left (x \right )}} \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\frac {2 \, x^{3} - x^{2} \log \left (x\right ) + x^{2} \log \left (2 \, \log \left (x\right ) - 1\right ) - x^{2} \log \left (\log \left (x\right )\right ) + x}{2 \, x - \log \left (x\right ) + \log \left (2 \, \log \left (x\right ) - 1\right ) - \log \left (\log \left (x\right )\right )} \]
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Time = 0.45 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=x^{2} + \frac {x}{2 \, x - \log \left (x\right ) + \log \left (2 \, \log \left (x\right ) - 1\right ) - \log \left (\log \left (x\right )\right )} \]
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Timed out. \[ \int \frac {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx=\int \frac {{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}^2\,\left (4\,x\,{\ln \left (\frac {1}{x}\right )}^2+2\,x\,\ln \left (\frac {1}{x}\right )\right )+\ln \left (\frac {1}{x}\right )\,\left (8\,x^3+1\right )+\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )\,\left (\ln \left (\frac {1}{x}\right )\,\left (8\,x^2+1\right )-\ln \left (x\right )\,\left (8\,x\,{\ln \left (\frac {1}{x}\right )}^2+4\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (\frac {1}{x}\right )}^2\,\left (16\,x^2+2\right )\right )-\ln \left (x\right )\,\left (\left (16\,x^2+2\right )\,{\ln \left (\frac {1}{x}\right )}^2+\left (8\,x^2+1\right )\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (x\right )}^2\,\left (4\,x\,{\ln \left (\frac {1}{x}\right )}^2+2\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (\frac {1}{x}\right )}^2\,\left (16\,x^3+2\right )-1}{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )\,\left (4\,x\,\ln \left (\frac {1}{x}\right )-\ln \left (x\right )\,\left (4\,{\ln \left (\frac {1}{x}\right )}^2+2\,\ln \left (\frac {1}{x}\right )\right )+8\,x\,{\ln \left (\frac {1}{x}\right )}^2\right )-\ln \left (x\right )\,\left (8\,x\,{\ln \left (\frac {1}{x}\right )}^2+4\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (x\right )}^2\,\left (2\,{\ln \left (\frac {1}{x}\right )}^2+\ln \left (\frac {1}{x}\right )\right )+4\,x^2\,\ln \left (\frac {1}{x}\right )+{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}^2\,\left (2\,{\ln \left (\frac {1}{x}\right )}^2+\ln \left (\frac {1}{x}\right )\right )+8\,x^2\,{\ln \left (\frac {1}{x}\right )}^2} \,d x \]
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