Integrand size = 221, antiderivative size = 33 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=e^{\frac {e^2 x}{\left (-3+e^x\right ) x-\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \]
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Time = 2.89 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {6820, 6838} \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=\exp \left (-\frac {e^2 x}{\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\left (3-e^x\right ) x}\right ) \]
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Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (2+\frac {e^2 x}{\left (-3+e^x\right ) x-\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}\right ) \left (-2-\log (x) \left (-2+\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \left (e^x x^2+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )\right )\right )}{\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \left (\left (-3+e^x\right ) x-\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )^2} \, dx \\ & = \exp \left (-\frac {e^2 x}{\left (3-e^x\right ) x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.92 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.58
\[{\mathrm e}^{\frac {{\mathrm e}^{2} x}{{\mathrm e}^{x} x -\ln \left (-2 \ln \left (\ln \left (2\right )\right )+2 \ln \left (x \right )-2 \ln \left (\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x^{2}}{\ln \left (x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right )\right )}{2}\right )-3 x}}\]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=e^{\left (-\frac {x e^{4}}{3 \, x e^{2} - x e^{\left (x + 2\right )} + e^{2} \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )}\right )} \]
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Timed out. \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=\int { \frac {{\left (x^{2} e^{\left (x + 2\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) + e^{2} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - 2 \, e^{2} \log \left (x\right ) + 2 \, e^{2}\right )} e^{\left (\frac {x e^{2}}{x e^{x} - 3 \, x - \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )}\right )}}{2 \, {\left (x e^{x} - 3 \, x\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )^{2} - {\left (x^{2} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{x} + 9 \, x^{2}\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )} \,d x } \]
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\[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=\int { \frac {{\left (x^{2} e^{\left (x + 2\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) + e^{2} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - 2 \, e^{2} \log \left (x\right ) + 2 \, e^{2}\right )} e^{\left (\frac {x e^{2}}{x e^{x} - 3 \, x - \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )}\right )}}{2 \, {\left (x e^{x} - 3 \, x\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )^{2} - {\left (x^{2} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{x} + 9 \, x^{2}\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )} \,d x } \]
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Time = 13.95 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx={\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^2}{3\,x+\ln \left (\ln \left (x^2\right )+\ln \left (\frac {1}{{\ln \left (x\right )}^2}\right )-2\,\ln \left (\ln \left (2\right )\right )\right )-x\,{\mathrm {e}}^x}} \]
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