Integrand size = 138, antiderivative size = 30 \[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=\left (-5+e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}}\right ) \log (2) \]
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\[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=\int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (5+e^x\right )^2 x \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx \\ & = \int \left (-\frac {5 e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (2) (x-\log (x))}{\left (5+e^x\right )^2 \log \left (\log \left (25 x^2\right )\right )}+\frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (2) \left (2 x-2 \log (x)+\log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )-x \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )+x^2 \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )-x \log (x) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (5+e^x\right ) x \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )}\right ) \, dx \\ & = \log (2) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (2 x-2 \log (x)+\log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )-x \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )+x^2 \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )-x \log (x) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (5+e^x\right ) x \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx-(5 \log (2)) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} (x-\log (x))}{\left (5+e^x\right )^2 \log \left (\log \left (25 x^2\right )\right )} \, dx \\ & = \log (2) \int \left (\frac {2 e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}}}{\left (5+e^x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )}-\frac {2 e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (x)}{\left (5+e^x\right ) x \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )}-\frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}}}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}+\frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}}}{\left (5+e^x\right ) x \log \left (\log \left (25 x^2\right )\right )}+\frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} x}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}-\frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}\right ) \, dx-(5 \log (2)) \int \left (\frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} x}{\left (5+e^x\right )^2 \log \left (\log \left (25 x^2\right )\right )}-\frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (x)}{\left (5+e^x\right )^2 \log \left (\log \left (25 x^2\right )\right )}\right ) \, dx \\ & = -\left (\log (2) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}}}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )} \, dx\right )+\log (2) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}}}{\left (5+e^x\right ) x \log \left (\log \left (25 x^2\right )\right )} \, dx+\log (2) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} x}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )} \, dx-\log (2) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )} \, dx+(2 \log (2)) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}}}{\left (5+e^x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx-(2 \log (2)) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (x)}{\left (5+e^x\right ) x \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx-(5 \log (2)) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} x}{\left (5+e^x\right )^2 \log \left (\log \left (25 x^2\right )\right )} \, dx+(5 \log (2)) \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \log (x)}{\left (5+e^x\right )^2 \log \left (\log \left (25 x^2\right )\right )} \, dx \\ \end{align*}
\[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=\int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.84 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93
\[\ln \left (2\right ) {\mathrm e}^{\frac {\ln \left (x \right )-x}{\left ({\mathrm e}^{x}+5\right ) \ln \left (2 \ln \left (x \right )+2 \ln \left (5\right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )}}\]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=e^{\left (-\frac {x - \log \left (x\right )}{{\left (e^{x} + 5\right )} \log \left (2 \, \log \left (5\right ) + 2 \, \log \left (x\right )\right )}\right )} \log \left (2\right ) \]
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Time = 33.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=e^{\frac {- x + \log {\left (x \right )}}{\left (e^{x} + 5\right ) \log {\left (2 \log {\left (x \right )} + \log {\left (25 \right )} \right )}}} \log {\left (2 \right )} \]
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Exception generated. \[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=\int { \frac {{\left (2 \, x e^{x} \log \left (2\right ) - {\left (x e^{x} \log \left (2\right ) \log \left (x\right ) - {\left (x^{2} - x + 1\right )} e^{x} \log \left (2\right ) + 5 \, {\left (x - 1\right )} \log \left (2\right )\right )} \log \left (25 \, x^{2}\right ) \log \left (\log \left (25 \, x^{2}\right )\right ) + 10 \, x \log \left (2\right ) - 2 \, {\left (e^{x} \log \left (2\right ) + 5 \, \log \left (2\right )\right )} \log \left (x\right )\right )} e^{\left (-\frac {x - \log \left (x\right )}{{\left (e^{x} + 5\right )} \log \left (\log \left (25 \, x^{2}\right )\right )}\right )}}{{\left (x e^{\left (2 \, x\right )} + 10 \, x e^{x} + 25 \, x\right )} \log \left (25 \, x^{2}\right ) \log \left (\log \left (25 \, x^{2}\right )\right )^{2}} \,d x } \]
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Time = 14.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {e^{\frac {-x+\log (x)}{\left (5+e^x\right ) \log \left (\log \left (25 x^2\right )\right )}} \left (10 x \log (2)+2 e^x x \log (2)+\left (-10 \log (2)-2 e^x \log (2)\right ) \log (x)+\left ((5-5 x) \log (2)+e^x \left (1-x+x^2\right ) \log (2)-e^x x \log (2) \log (x)\right ) \log \left (25 x^2\right ) \log \left (\log \left (25 x^2\right )\right )\right )}{\left (25 x+10 e^x x+e^{2 x} x\right ) \log \left (25 x^2\right ) \log ^2\left (\log \left (25 x^2\right )\right )} \, dx=x^{\frac {1}{5\,\ln \left (\ln \left (25\,x^2\right )\right )+\ln \left (\ln \left (25\,x^2\right )\right )\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {x}{5\,\ln \left (\ln \left (25\,x^2\right )\right )+\ln \left (\ln \left (25\,x^2\right )\right )\,{\mathrm {e}}^x}}\,\ln \left (2\right ) \]
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