Integrand size = 107, antiderivative size = 21 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=(4-x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \]
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\[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=\int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx \\ & = \int \frac {2 \left (4-x-(-2+x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right )}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx \\ & = 2 \int \frac {4-x-(-2+x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx \\ & = 2 \int \left (\frac {4-x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}-(-2+x) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \, dx \\ & = 2 \int \frac {4-x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx-2 \int (-2+x) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx \\ & = 2 \int \left (\frac {4-x}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}+\frac {(-4+x) \log (3)}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}\right ) \, dx-2 \int \left (-2 \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )+x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \, dx \\ & = -\left (2 \int x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\right )+4 \int \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx+\log (3) \int \frac {-4+x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx+\int \frac {4-x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx \\ & = -\left (2 \int x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\right )+4 \int \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx+\log (3) \int \left (-\frac {4}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}+\frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}\right ) \, dx+\int \left (\frac {4}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}-\frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx \\ & = -\left (2 \int x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\right )+4 \int \frac {1}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx+4 \int \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx+\log (3) \int \frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx-(4 \log (3)) \int \frac {1}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx-\int \frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-\left ((-4+x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.22 (sec) , antiderivative size = 397, normalized size of antiderivative = 18.90
\[\left (-x^{2}+4 x \right ) \ln \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )+x^{2} \ln \left (\ln \left (-\ln \left (\ln \left (x \right )\right )\right )\right )-4 x \ln \left (\ln \left (-\ln \left (\ln \left (x \right )\right )\right )\right )+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{3}}{2}-2 i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )+2 i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}+2 i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}-2 i \pi x \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{3}\]
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-{\left (x^{2} - 4 \, x\right )} \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {1}{\log \left (x\right )}\right )\right ) + 2}{\log \left (\log \left (\frac {1}{\log \left (x\right )}\right )\right )}\right ) \]
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Time = 1.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=\left (- x^{2} + 4 x\right ) \log {\left (\frac {\log {\left (3 \right )} \log {\left (\log {\left (\frac {1}{\log {\left (x \right )}} \right )} \right )} + 2}{\log {\left (\log {\left (\frac {1}{\log {\left (x \right )}} \right )} \right )}} \right )} \]
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Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-{\left (x^{2} - 4 \, x\right )} \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) + {\left (x^{2} - 4 \, x\right )} \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
Time = 1.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.57 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-x^{2} \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) + x^{2} \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) + 4 \, x \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) - 4 \, x \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) \]
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Time = 15.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-x\,\ln \left (\frac {\ln \left (\ln \left (\frac {1}{\ln \left (x\right )}\right )\right )\,\ln \left (3\right )+2}{\ln \left (\ln \left (\frac {1}{\ln \left (x\right )}\right )\right )}\right )\,\left (x-4\right ) \]
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