Integrand size = 125, antiderivative size = 20 \[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx=1+e^{\left (1+\frac {1}{x+\frac {5}{\log (2 x)}}\right )^2} \]
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\[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx=\int \frac {\exp \left (\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}\right ) \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{x (5+x \log (2 x))^3} \, dx \\ & = \int \left (-\frac {2 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) (1+x)}{x^3}-\frac {50 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) (-5+x)}{x^3 (5+x \log (2 x))^3}+\frac {10 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) \left (-15-4 x+x^2\right )}{x^3 (5+x \log (2 x))^2}+\frac {10 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) (3+2 x)}{x^3 (5+x \log (2 x))}\right ) \, dx \\ & = -\left (2 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) (1+x)}{x^3} \, dx\right )+10 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) \left (-15-4 x+x^2\right )}{x^3 (5+x \log (2 x))^2} \, dx+10 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) (3+2 x)}{x^3 (5+x \log (2 x))} \, dx-50 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right ) (-5+x)}{x^3 (5+x \log (2 x))^3} \, dx \\ & = -\left (2 \int \left (\frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3}+\frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2}\right ) \, dx\right )+10 \int \left (-\frac {15 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3 (5+x \log (2 x))^2}-\frac {4 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2 (5+x \log (2 x))^2}+\frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x (5+x \log (2 x))^2}\right ) \, dx+10 \int \left (\frac {3 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3 (5+x \log (2 x))}+\frac {2 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2 (5+x \log (2 x))}\right ) \, dx-50 \int \left (-\frac {5 \exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3 (5+x \log (2 x))^3}+\frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2 (5+x \log (2 x))^3}\right ) \, dx \\ & = -\left (2 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3} \, dx\right )-2 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2} \, dx+10 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x (5+x \log (2 x))^2} \, dx+20 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2 (5+x \log (2 x))} \, dx+30 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3 (5+x \log (2 x))} \, dx-40 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2 (5+x \log (2 x))^2} \, dx-50 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^2 (5+x \log (2 x))^3} \, dx-150 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3 (5+x \log (2 x))^2} \, dx+250 \int \frac {\exp \left (\frac {(5+\log (2 x)+x \log (2 x))^2}{(5+x \log (2 x))^2}\right )}{x^3 (5+x \log (2 x))^3} \, dx \\ \end{align*}
\[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx=\int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx \]
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Time = 6.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35
method | result | size |
risch | \({\mathrm e}^{\frac {\left (x \ln \left (2 x \right )+\ln \left (2 x \right )+5\right )^{2}}{\left (x \ln \left (2 x \right )+5\right )^{2}}}\) | \(27\) |
parallelrisch | \({\mathrm e}^{\frac {\left (x^{2}+2 x +1\right ) \ln \left (2 x \right )^{2}+\left (10 x +10\right ) \ln \left (2 x \right )+25}{x^{2} \ln \left (2 x \right )^{2}+10 x \ln \left (2 x \right )+25}}\) | \(51\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx=e^{\left (\frac {{\left (x^{2} + 2 \, x + 1\right )} \log \left (2 \, x\right )^{2} + 10 \, {\left (x + 1\right )} \log \left (2 \, x\right ) + 25}{x^{2} \log \left (2 \, x\right )^{2} + 10 \, x \log \left (2 \, x\right ) + 25}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).
Time = 0.42 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.40 \[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx=e^{\frac {\left (10 x + 10\right ) \log {\left (2 x \right )} + \left (x^{2} + 2 x + 1\right ) \log {\left (2 x \right )}^{2} + 25}{x^{2} \log {\left (2 x \right )}^{2} + 10 x \log {\left (2 x \right )} + 25}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (19) = 38\).
Time = 0.66 (sec) , antiderivative size = 168, normalized size of antiderivative = 8.40 \[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx=e^{\left (\frac {\log \left (2\right )^{2}}{x^{2} \log \left (2\right )^{2} + x^{2} \log \left (x\right )^{2} + 10 \, x \log \left (2\right ) + 2 \, {\left (x^{2} \log \left (2\right ) + 5 \, x\right )} \log \left (x\right ) + 25} + \frac {2 \, \log \left (2\right ) \log \left (x\right )}{x^{2} \log \left (2\right )^{2} + x^{2} \log \left (x\right )^{2} + 10 \, x \log \left (2\right ) + 2 \, {\left (x^{2} \log \left (2\right ) + 5 \, x\right )} \log \left (x\right ) + 25} + \frac {\log \left (x\right )^{2}}{x^{2} \log \left (2\right )^{2} + x^{2} \log \left (x\right )^{2} + 10 \, x \log \left (2\right ) + 2 \, {\left (x^{2} \log \left (2\right ) + 5 \, x\right )} \log \left (x\right ) + 25} + \frac {2 \, \log \left (2\right )}{x \log \left (2\right ) + x \log \left (x\right ) + 5} + \frac {2 \, \log \left (x\right )}{x \log \left (2\right ) + x \log \left (x\right ) + 5} + 1\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (19) = 38\).
Time = 2.37 (sec) , antiderivative size = 169, normalized size of antiderivative = 8.45 \[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx=e^{\left (\frac {x^{2} \log \left (2 \, x\right )^{2}}{x^{2} \log \left (2 \, x\right )^{2} + 10 \, x \log \left (2 \, x\right ) + 25} + \frac {2 \, x \log \left (2 \, x\right )^{2}}{x^{2} \log \left (2 \, x\right )^{2} + 10 \, x \log \left (2 \, x\right ) + 25} + \frac {10 \, x \log \left (2 \, x\right )}{x^{2} \log \left (2 \, x\right )^{2} + 10 \, x \log \left (2 \, x\right ) + 25} + \frac {\log \left (2 \, x\right )^{2}}{x^{2} \log \left (2 \, x\right )^{2} + 10 \, x \log \left (2 \, x\right ) + 25} + \frac {10 \, \log \left (2 \, x\right )}{x^{2} \log \left (2 \, x\right )^{2} + 10 \, x \log \left (2 \, x\right ) + 25} + \frac {25}{x^{2} \log \left (2 \, x\right )^{2} + 10 \, x \log \left (2 \, x\right ) + 25}\right )} \]
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Time = 9.57 (sec) , antiderivative size = 429, normalized size of antiderivative = 21.45 \[ \int \frac {e^{\frac {25+(10+10 x) \log (2 x)+\left (1+2 x+x^2\right ) \log ^2(2 x)}{25+10 x \log (2 x)+x^2 \log ^2(2 x)}} \left (50+(10+10 x) \log (2 x)-10 x \log ^2(2 x)+\left (-2 x-2 x^2\right ) \log ^3(2 x)\right )}{125 x+75 x^2 \log (2 x)+15 x^3 \log ^2(2 x)+x^4 \log ^3(2 x)} \, dx={1024}^{\frac {x+1}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,x^{\frac {2\,\left (5\,x+\ln \left (2\right )+2\,x\,\ln \left (2\right )+x^2\,\ln \left (2\right )+5\right )}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,{\mathrm {e}}^{\frac {x^2\,{\ln \left (2\right )}^2}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,{\mathrm {e}}^{\frac {2\,x\,{\ln \left (x\right )}^2}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,{\mathrm {e}}^{\frac {{\ln \left (2\right )}^2}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,{\mathrm {e}}^{\frac {25}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,{\mathrm {e}}^{\frac {x^2\,{\ln \left (x\right )}^2}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,{\mathrm {e}}^{\frac {{\ln \left (x\right )}^2}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}}\,{\mathrm {e}}^{\frac {2\,x\,{\ln \left (2\right )}^2}{x^2\,{\ln \left (x\right )}^2+2\,\ln \left (2\right )\,x^2\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^2+10\,x\,\ln \left (x\right )+10\,\ln \left (2\right )\,x+25}} \]
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