Integrand size = 170, antiderivative size = 26 \[ \int \frac {e^{2 e^{-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )}-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )} \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx=e^{2 e^{-x^3 \left (-x+\log \left (\frac {45}{x}+\log (5)\right )\right )^2}} \]
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {e^{2 e^{-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )}-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )} \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx=e^{2 e^{-x^5-x^3 \log ^2\left (\frac {45}{x}+\log (5)\right )} \left (\frac {45}{x}+\log (5)\right )^{2 x^4}} \]
Integrate[(E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*L og[5])/x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5] )/x]^2)*(-180*x^3 - 450*x^4 - 10*x^5*Log[5] + (180*x^2 + 720*x^3 + 16*x^4* Log[5])*Log[(45 + x*Log[5])/x] + (-270*x^2 - 6*x^3*Log[5])*Log[(45 + x*Log [5])/x]^2))/(45 + x*Log[5]),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-10 x^5 \log (5)-450 x^4-180 x^3+\left (-6 x^3 \log (5)-270 x^2\right ) \log ^2\left (\frac {x \log (5)+45}{x}\right )+\left (16 x^4 \log (5)+720 x^3+180 x^2\right ) \log \left (\frac {x \log (5)+45}{x}\right )\right ) \exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {x \log (5)+45}{x}\right )-x^3 \log ^2\left (\frac {x \log (5)+45}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {x \log (5)+45}{x}\right )-x^3 \log ^2\left (\frac {x \log (5)+45}{x}\right )\right )}{x \log (5)+45} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 x^2 \left (x-\log \left (\frac {45}{x}+\log (5)\right )\right ) \left (-5 x^2 \log (5)-225 x+3 x \log (5) \log \left (\frac {45}{x}+\log (5)\right )+135 \log \left (\frac {45}{x}+\log (5)\right )-90\right ) \exp \left (2 \exp \left (-x^5+2 x^4 \log \left (\frac {x \log (5)+45}{x}\right )-x^3 \log ^2\left (\frac {x \log (5)+45}{x}\right )\right )-x^5+2 x^4 \log \left (\frac {x \log (5)+45}{x}\right )-x^3 \log ^2\left (\frac {x \log (5)+45}{x}\right )\right )}{x \log (5)+45}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\exp \left (2 e^{-x^5-\log ^2\left (\frac {\log (5) x+45}{x}\right ) x^3} \left (\frac {\log (5) x+45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\frac {\log (5) x+45}{x}\right )\right ) x^2 \left (\frac {\log (5) x+45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \log (5) x^2-3 \log (5) \log \left (\log (5)+\frac {45}{x}\right ) x+225 x-135 \log \left (\log (5)+\frac {45}{x}\right )+90\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^5-\log ^2\left (\frac {\log (5) x+45}{x}\right ) x^3} \left (\frac {\log (5) x+45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\frac {\log (5) x+45}{x}\right )\right ) x^2 \left (\frac {\log (5) x+45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \log (5) x^2-3 \log (5) \log \left (\log (5)+\frac {45}{x}\right ) x+225 x-135 \log \left (\log (5)+\frac {45}{x}\right )+90\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^5-\log ^2\left (\frac {\log (5) x+45}{x}\right ) x^3} \left (\frac {\log (5) x+45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\frac {\log (5) x+45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \log (5) x^2-3 \log (5) \log \left (\log (5)+\frac {45}{x}\right ) x+225 x-135 \log \left (\log (5)+\frac {45}{x}\right )+90\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^5-\log ^2\left (\frac {\log (5) x+45}{x}\right ) x^3} \left (\frac {\log (5) x+45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\frac {\log (5) x+45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^5-\log ^2\left (\frac {\log (5) x+45}{x}\right ) x^3} \left (\frac {\log (5) x+45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\frac {\log (5) x+45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^5-\log ^2\left (\frac {\log (5) x+45}{x}\right ) x^3} \left (\frac {\log (5) x+45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\frac {\log (5) x+45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (3 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \log ^2\left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}+\frac {5 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^3 \left (\log (5) x^2+45 x+18\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}-\frac {2 \exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (4 \log (5) x^2+180 x+45\right ) \log \left (\log (5)+\frac {45}{x}\right ) \left (\log (5)+\frac {45}{x}\right )^{2 x^4}}{\log (5) x+45}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 e^{-x^3 \left (x^2+\log ^2\left (\log (5)+\frac {45}{x}\right )\right )} \left (\log (5)+\frac {45}{x}\right )^{2 x^4}-x^5-x^3 \log ^2\left (\log (5)+\frac {45}{x}\right )\right ) x^2 \left (\log (5)+\frac {45}{x}\right )^{2 x^4} \left (x-\log \left (\log (5)+\frac {45}{x}\right )\right ) \left (5 \left (\log (5) x^2+45 x+18\right )-3 (\log (5) x+45) \log \left (\log (5)+\frac {45}{x}\right )\right )}{\log (5) x+45}dx\) |
Int[(E^(2*E^(-x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5]) /x]^2) - x^5 + 2*x^4*Log[(45 + x*Log[5])/x] - x^3*Log[(45 + x*Log[5])/x]^2 )*(-180*x^3 - 450*x^4 - 10*x^5*Log[5] + (180*x^2 + 720*x^3 + 16*x^4*Log[5] )*Log[(45 + x*Log[5])/x] + (-270*x^2 - 6*x^3*Log[5])*Log[(45 + x*Log[5])/x ]^2))/(45 + x*Log[5]),x]
3.14.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 28.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69
| method | result | size |
| parallelrisch | \({\mathrm e}^{2 \,{\mathrm e}^{-x^{3} \ln \left (\frac {x \ln \left (5\right )+45}{x}\right )^{2}+2 x^{4} \ln \left (\frac {x \ln \left (5\right )+45}{x}\right )-x^{5}}}\) | \(44\) |
| risch | \({\mathrm e}^{2 \left (\frac {x \ln \left (5\right )+45}{x}\right )^{2 x^{4}} {\mathrm e}^{-x^{3} \left (\ln \left (\frac {x \ln \left (5\right )+45}{x}\right )^{2}+x^{2}\right )}}\) | \(45\) |
int(((-6*x^3*ln(5)-270*x^2)*ln((x*ln(5)+45)/x)^2+(16*x^4*ln(5)+720*x^3+180 *x^2)*ln((x*ln(5)+45)/x)-10*x^5*ln(5)-450*x^4-180*x^3)*exp(2/exp(x^3*ln((x *ln(5)+45)/x)^2-2*x^4*ln((x*ln(5)+45)/x)+x^5))/(x*ln(5)+45)/exp(x^3*ln((x* ln(5)+45)/x)^2-2*x^4*ln((x*ln(5)+45)/x)+x^5),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{2 e^{-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )}-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )} \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx=e^{\left (2 \, e^{\left (-x^{5} + 2 \, x^{4} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right ) - x^{3} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right )^{2}\right )}\right )} \]
integrate(((-6*x^3*log(5)-270*x^2)*log((x*log(5)+45)/x)^2+(16*x^4*log(5)+7 20*x^3+180*x^2)*log((x*log(5)+45)/x)-10*x^5*log(5)-450*x^4-180*x^3)*exp(2/ exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5))/(x*log(5)+ 45)/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5),x, algo rithm=\
Time = 2.67 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {e^{2 e^{-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )}-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )} \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx=e^{2 e^{- x^{5} + 2 x^{4} \log {\left (\frac {x \log {\left (5 \right )} + 45}{x} \right )} - x^{3} \log {\left (\frac {x \log {\left (5 \right )} + 45}{x} \right )}^{2}}} \]
integrate(((-6*x**3*ln(5)-270*x**2)*ln((x*ln(5)+45)/x)**2+(16*x**4*ln(5)+7 20*x**3+180*x**2)*ln((x*ln(5)+45)/x)-10*x**5*ln(5)-450*x**4-180*x**3)*exp( 2/exp(x**3*ln((x*ln(5)+45)/x)**2-2*x**4*ln((x*ln(5)+45)/x)+x**5))/(x*ln(5) +45)/exp(x**3*ln((x*ln(5)+45)/x)**2-2*x**4*ln((x*ln(5)+45)/x)+x**5),x)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 1.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int \frac {e^{2 e^{-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )}-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )} \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx=e^{\left (2 \, e^{\left (-x^{5} + 2 \, x^{4} \log \left (x \log \left (5\right ) + 45\right ) - x^{3} \log \left (x \log \left (5\right ) + 45\right )^{2} - 2 \, x^{4} \log \left (x\right ) + 2 \, x^{3} \log \left (x \log \left (5\right ) + 45\right ) \log \left (x\right ) - x^{3} \log \left (x\right )^{2}\right )}\right )} \]
integrate(((-6*x^3*log(5)-270*x^2)*log((x*log(5)+45)/x)^2+(16*x^4*log(5)+7 20*x^3+180*x^2)*log((x*log(5)+45)/x)-10*x^5*log(5)-450*x^4-180*x^3)*exp(2/ exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5))/(x*log(5)+ 45)/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5),x, algo rithm=\
e^(2*e^(-x^5 + 2*x^4*log(x*log(5) + 45) - x^3*log(x*log(5) + 45)^2 - 2*x^4 *log(x) + 2*x^3*log(x*log(5) + 45)*log(x) - x^3*log(x)^2))
\[ \int \frac {e^{2 e^{-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )}-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )} \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx=\int { -\frac {2 \, {\left (5 \, x^{5} \log \left (5\right ) + 225 \, x^{4} + 90 \, x^{3} + 3 \, {\left (x^{3} \log \left (5\right ) + 45 \, x^{2}\right )} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right )^{2} - 2 \, {\left (4 \, x^{4} \log \left (5\right ) + 180 \, x^{3} + 45 \, x^{2}\right )} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right )\right )} e^{\left (-x^{5} + 2 \, x^{4} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right ) - x^{3} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right )^{2} + 2 \, e^{\left (-x^{5} + 2 \, x^{4} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right ) - x^{3} \log \left (\frac {x \log \left (5\right ) + 45}{x}\right )^{2}\right )}\right )}}{x \log \left (5\right ) + 45} \,d x } \]
integrate(((-6*x^3*log(5)-270*x^2)*log((x*log(5)+45)/x)^2+(16*x^4*log(5)+7 20*x^3+180*x^2)*log((x*log(5)+45)/x)-10*x^5*log(5)-450*x^4-180*x^3)*exp(2/ exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5))/(x*log(5)+ 45)/exp(x^3*log((x*log(5)+45)/x)^2-2*x^4*log((x*log(5)+45)/x)+x^5),x, algo rithm=\
integrate(-2*(5*x^5*log(5) + 225*x^4 + 90*x^3 + 3*(x^3*log(5) + 45*x^2)*lo g((x*log(5) + 45)/x)^2 - 2*(4*x^4*log(5) + 180*x^3 + 45*x^2)*log((x*log(5) + 45)/x))*e^(-x^5 + 2*x^4*log((x*log(5) + 45)/x) - x^3*log((x*log(5) + 45 )/x)^2 + 2*e^(-x^5 + 2*x^4*log((x*log(5) + 45)/x) - x^3*log((x*log(5) + 45 )/x)^2))/(x*log(5) + 45), x)
Time = 15.46 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {e^{2 e^{-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )}-x^5+2 x^4 \log \left (\frac {45+x \log (5)}{x}\right )-x^3 \log ^2\left (\frac {45+x \log (5)}{x}\right )} \left (-180 x^3-450 x^4-10 x^5 \log (5)+\left (180 x^2+720 x^3+16 x^4 \log (5)\right ) \log \left (\frac {45+x \log (5)}{x}\right )+\left (-270 x^2-6 x^3 \log (5)\right ) \log ^2\left (\frac {45+x \log (5)}{x}\right )\right )}{45+x \log (5)} \, dx={\mathrm {e}}^{2\,{\mathrm {e}}^{-x^3\,{\ln \left (\frac {x\,\ln \left (5\right )+45}{x}\right )}^2}\,{\mathrm {e}}^{-x^5}\,{\left (\ln \left (5\right )+\frac {45}{x}\right )}^{2\,x^4}} \]
int(-(exp(2*exp(2*x^4*log((x*log(5) + 45)/x) - x^5 - x^3*log((x*log(5) + 4 5)/x)^2))*exp(2*x^4*log((x*log(5) + 45)/x) - x^5 - x^3*log((x*log(5) + 45) /x)^2)*(log((x*log(5) + 45)/x)^2*(6*x^3*log(5) + 270*x^2) - log((x*log(5) + 45)/x)*(16*x^4*log(5) + 180*x^2 + 720*x^3) + 10*x^5*log(5) + 180*x^3 + 4 50*x^4))/(x*log(5) + 45),x)