Integrand size = 120, antiderivative size = 27 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\frac {3}{\frac {2+x}{x \log (x)}-\log \left (\log \left (\frac {3}{x}\right )\right )}} \]
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\[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left (-3 x \log ^2(x)+3 \log \left (\frac {3}{x}\right ) (2+x+2 \log (x))\right )}{\log \left (\frac {3}{x}\right ) \left (2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx \\ & = \int \left (\frac {6 x^{\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}}}{\left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2}+\frac {3 x^{1+\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}}}{\left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2}+\frac {6 x^{\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \log (x)}{\left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2}-\frac {3 x^{1+\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \log ^2(x)}{\log \left (\frac {3}{x}\right ) \left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2}\right ) \, dx \\ & = 3 \int \frac {x^{1+\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}}}{\left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx-3 \int \frac {x^{1+\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \log ^2(x)}{\log \left (\frac {3}{x}\right ) \left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx+6 \int \frac {x^{\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}}}{\left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx+6 \int \frac {x^{\frac {3 x}{2+x-x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \log (x)}{\left (-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )\right )^2} \, dx \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \]
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Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[x^{-\frac {3 x}{x \ln \left (x \right ) \ln \left (-\ln \left (x \right )+\ln \left (3\right )\right )-x -2}}\]
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Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\left (-\frac {3 \, {\left (x \log \left (3\right ) - x \log \left (\frac {3}{x}\right )\right )}}{{\left (x \log \left (3\right ) - x \log \left (\frac {3}{x}\right )\right )} \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}\right )} \]
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Time = 2.56 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{- \frac {3 x \log {\left (x \right )}}{x \log {\left (x \right )} \log {\left (- \log {\left (x \right )} + \log {\left (3 \right )} \right )} - x - 2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).
Time = 0.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=e^{\left (-\frac {6 \, \log \left (x\right )}{x \log \left (x\right )^{2} \log \left (\log \left (3\right ) - \log \left (x\right )\right )^{2} - 2 \, {\left (x \log \left (\log \left (3\right ) - \log \left (x\right )\right ) + \log \left (\log \left (3\right ) - \log \left (x\right )\right )\right )} \log \left (x\right ) + x + 2} - \frac {3 \, \log \left (x\right )}{\log \left (x\right ) \log \left (\log \left (3\right ) - \log \left (x\right )\right ) - 1}\right )} \]
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Time = 0.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx=\frac {1}{x^{\frac {3 \, x}{x \log \left (x\right ) \log \left (\log \left (\frac {3}{x}\right )\right ) - x - 2}}} \]
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Time = 12.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^{-\frac {3 x}{-2-x+x \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )}} \left ((6+3 x) \log \left (\frac {3}{x}\right )+6 \log \left (\frac {3}{x}\right ) \log (x)-3 x \log ^2(x)\right )}{\left (4+4 x+x^2\right ) \log \left (\frac {3}{x}\right )+\left (-4 x-2 x^2\right ) \log \left (\frac {3}{x}\right ) \log (x) \log \left (\log \left (\frac {3}{x}\right )\right )+x^2 \log \left (\frac {3}{x}\right ) \log ^2(x) \log ^2\left (\log \left (\frac {3}{x}\right )\right )} \, dx={\mathrm {e}}^{\frac {3\,x\,\ln \left (x\right )}{x-x\,\ln \left (\ln \left (\frac {1}{x}\right )+\ln \left (3\right )\right )\,\ln \left (x\right )+2}} \]
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