Integrand size = 204, antiderivative size = 32 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=-\frac {x}{e^{e^x}+x}+x \left (1+\frac {x}{3}+\log (x)\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right ) \]
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\[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{3} \left (\frac {3 e^{e^x} \left (-1+e^x x\right )}{\left (e^{e^x}+x\right )^2}+\frac {3+x+3 \log (x)}{\log (x) \log \left (\frac {\log (x)}{2}\right )}+(6+2 x+3 \log (x)) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )\right ) \, dx \\ & = \frac {1}{3} \int \left (\frac {3 e^{e^x} \left (-1+e^x x\right )}{\left (e^{e^x}+x\right )^2}+\frac {3+x+3 \log (x)}{\log (x) \log \left (\frac {\log (x)}{2}\right )}+(6+2 x+3 \log (x)) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )\right ) \, dx \\ & = \frac {1}{3} \int \frac {3+x+3 \log (x)}{\log (x) \log \left (\frac {\log (x)}{2}\right )} \, dx+\frac {1}{3} \int (6+2 x+3 \log (x)) \log \left (\log \left (\frac {\log (x)}{2}\right )\right ) \, dx+\int \frac {e^{e^x} \left (-1+e^x x\right )}{\left (e^{e^x}+x\right )^2} \, dx \\ & = \frac {1}{3} \int \left (\frac {3}{\log \left (\frac {\log (x)}{2}\right )}+\frac {3}{\log (x) \log \left (\frac {\log (x)}{2}\right )}+\frac {x}{\log (x) \log \left (\frac {\log (x)}{2}\right )}\right ) \, dx+\frac {1}{3} \int \left (6 \log \left (\log \left (\frac {\log (x)}{2}\right )\right )+2 x \log \left (\log \left (\frac {\log (x)}{2}\right )\right )+3 \log (x) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )\right ) \, dx+\text {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\frac {e^{e^x}}{x}\right ) \\ & = -\frac {1}{1+\frac {e^{e^x}}{x}}+\frac {1}{3} \int \frac {x}{\log (x) \log \left (\frac {\log (x)}{2}\right )} \, dx+\frac {2}{3} \int x \log \left (\log \left (\frac {\log (x)}{2}\right )\right ) \, dx+2 \int \log \left (\log \left (\frac {\log (x)}{2}\right )\right ) \, dx+\int \frac {1}{\log \left (\frac {\log (x)}{2}\right )} \, dx+\int \frac {1}{\log (x) \log \left (\frac {\log (x)}{2}\right )} \, dx+\int \log (x) \log \left (\log \left (\frac {\log (x)}{2}\right )\right ) \, dx \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\frac {1}{3} \left (-\frac {3 x}{e^{e^x}+x}+x (3+x+3 \log (x)) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )\right ) \]
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Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94
\[\left (x \ln \left (x \right )+\frac {x^{2}}{3}+x \right ) \ln \left (\ln \left (\frac {\ln \left (x \right )}{2}\right )\right )-\frac {x}{x +{\mathrm e}^{{\mathrm e}^{x}}}\]
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Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\frac {{\left (x^{3} + 3 \, x^{2} \log \left (x\right ) + 3 \, x^{2} + {\left (x^{2} + 3 \, x \log \left (x\right ) + 3 \, x\right )} e^{\left (e^{x}\right )}\right )} \log \left (\log \left (\frac {1}{2} \, \log \left (x\right )\right )\right ) - 3 \, x}{3 \, {\left (x + e^{\left (e^{x}\right )}\right )}} \]
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Time = 1.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=- \frac {x}{x + e^{e^{x}}} + \left (\frac {x^{2}}{3} + x \log {\left (x \right )} + x\right ) \log {\left (\log {\left (\frac {\log {\left (x \right )}}{2} \right )} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\frac {{\left (x^{3} + 3 \, x^{2} \log \left (x\right ) + 3 \, x^{2} + {\left (x^{2} + 3 \, x \log \left (x\right ) + 3 \, x\right )} e^{\left (e^{x}\right )}\right )} \log \left (-\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) - 3 \, x}{3 \, {\left (x + e^{\left (e^{x}\right )}\right )}} \]
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Timed out. \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\text {Timed out} \]
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Time = 15.53 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\ln \left (\ln \left (\frac {\ln \left (x\right )}{2}\right )\right )\,\left (\frac {x^3+6\,x^2}{3\,x}-x+x\,\ln \left (x\right )\right )-\frac {x}{x+{\mathrm {e}}^{{\mathrm {e}}^x}} \]
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