Integrand size = 189, antiderivative size = 32 \[ \int \frac {e^{\frac {\left (2500+250 x-400 x^2+50 x^3\right ) \log ^2(x)}{-5-x+\left (1875-750 x+75 x^2\right ) \log ^2(x)}} \left (\left (-50000-15000 x+7000 x^2+600 x^3-200 x^4\right ) \log (x)+\left (2500 x+8000 x^2-700 x^3-200 x^4\right ) \log ^2(x)+\left (4687500 x-3750000 x^2+1125000 x^3-150000 x^4+7500 x^5\right ) \log ^4(x)\right )}{25 x+10 x^2+x^3+\left (-18750 x+3750 x^2+750 x^3-150 x^4\right ) \log ^2(x)+\left (3515625 x-2812500 x^2+843750 x^3-112500 x^4+5625 x^5\right ) \log ^4(x)} \, dx=2 e^{\frac {4+2 x}{3-\frac {5+x}{25 (5-x)^2 \log ^2(x)}}} \]
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {\left (2500+250 x-400 x^2+50 x^3\right ) \log ^2(x)}{-5-x+\left (1875-750 x+75 x^2\right ) \log ^2(x)}} \left (\left (-50000-15000 x+7000 x^2+600 x^3-200 x^4\right ) \log (x)+\left (2500 x+8000 x^2-700 x^3-200 x^4\right ) \log ^2(x)+\left (4687500 x-3750000 x^2+1125000 x^3-150000 x^4+7500 x^5\right ) \log ^4(x)\right )}{25 x+10 x^2+x^3+\left (-18750 x+3750 x^2+750 x^3-150 x^4\right ) \log ^2(x)+\left (3515625 x-2812500 x^2+843750 x^3-112500 x^4+5625 x^5\right ) \log ^4(x)} \, dx=2 e^{-\frac {50 (-5+x)^2 (2+x) \log ^2(x)}{5+x-75 (-5+x)^2 \log ^2(x)}} \]
Integrate[(E^(((2500 + 250*x - 400*x^2 + 50*x^3)*Log[x]^2)/(-5 - x + (1875 - 750*x + 75*x^2)*Log[x]^2))*((-50000 - 15000*x + 7000*x^2 + 600*x^3 - 20 0*x^4)*Log[x] + (2500*x + 8000*x^2 - 700*x^3 - 200*x^4)*Log[x]^2 + (468750 0*x - 3750000*x^2 + 1125000*x^3 - 150000*x^4 + 7500*x^5)*Log[x]^4))/(25*x + 10*x^2 + x^3 + (-18750*x + 3750*x^2 + 750*x^3 - 150*x^4)*Log[x]^2 + (351 5625*x - 2812500*x^2 + 843750*x^3 - 112500*x^4 + 5625*x^5)*Log[x]^4),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.
| \(\displaystyle \int \frac {\left (\left (-200 x^4-700 x^3+8000 x^2+2500 x\right ) \log ^2(x)+\left (-200 x^4+600 x^3+7000 x^2-15000 x-50000\right ) \log (x)+\left (7500 x^5-150000 x^4+1125000 x^3-3750000 x^2+4687500 x\right ) \log ^4(x)\right ) \exp \left (\frac {\left (50 x^3-400 x^2+250 x+2500\right ) \log ^2(x)}{\left (75 x^2-750 x+1875\right ) \log ^2(x)-x-5}\right )}{x^3+10 x^2+\left (-150 x^4+750 x^3+3750 x^2-18750 x\right ) \log ^2(x)+\left (5625 x^5-112500 x^4+843750 x^3-2812500 x^2+3515625 x\right ) \log ^4(x)+25 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {100 (5-x) \log (x) \left (x \left (2 x^2+17 x+5\right ) \log (x)+2 \left (x^3+2 x^2-25 x-50\right )-75 (x-5)^3 x \log ^3(x)\right ) \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{x-75 (x-5)^2 \log ^2(x)+5}\right )}{x \left (x-75 (x-5)^2 \log ^2(x)+5\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 100 \int -\frac {\exp \left (-\frac {50 (5-x)^2 (x+2) \log ^2(x)}{-75 (5-x)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (-75 (5-x)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)+2 \left (-x^3-2 x^2+25 x+50\right )\right )}{x \left (-75 (5-x)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (5-x)^2 (x+2) \log ^2(x)}{-75 (5-x)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (-75 (5-x)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)+2 \left (-x^3-2 x^2+25 x+50\right )\right )}{x \left (-75 (5-x)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (5-x)^2 (x+2) \log ^2(x)}{-75 (5-x)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (5-x)^2 (x+2) \log ^2(x)}{-75 (5-x)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (5-x)^2 (x+2) \log ^2(x)}{-75 (5-x)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -100 \int \frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (5-x) \log (x) \left (75 (x-5)^3 x \log ^3(x)-x \left (2 x^2+17 x+5\right ) \log (x)-2 \left (x^3+2 x^2-25 x-50\right )\right )}{x \left (-75 (x-5)^2 \log ^2(x)+x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -100 \int \left (\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) (17 x+55)}{75 (x-5) \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )}-\frac {1}{75} \exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right )+\frac {\exp \left (-\frac {50 (x-5)^2 (x+2) \log ^2(x)}{-75 (x-5)^2 \log ^2(x)+x+5}\right ) \left (x^2+7 x+10\right ) \left (150 \log (x) x^3-2250 \log (x) x^2+x^2+11250 \log (x) x+15 x-18750 \log (x)\right )}{75 (x-5) x \left (75 x^2 \log ^2(x)-750 x \log ^2(x)+1875 \log ^2(x)-x-5\right )^2}\right )dx\) |
Int[(E^(((2500 + 250*x - 400*x^2 + 50*x^3)*Log[x]^2)/(-5 - x + (1875 - 750 *x + 75*x^2)*Log[x]^2))*((-50000 - 15000*x + 7000*x^2 + 600*x^3 - 200*x^4) *Log[x] + (2500*x + 8000*x^2 - 700*x^3 - 200*x^4)*Log[x]^2 + (4687500*x - 3750000*x^2 + 1125000*x^3 - 150000*x^4 + 7500*x^5)*Log[x]^4))/(25*x + 10*x ^2 + x^3 + (-18750*x + 3750*x^2 + 750*x^3 - 150*x^4)*Log[x]^2 + (3515625*x - 2812500*x^2 + 843750*x^3 - 112500*x^4 + 5625*x^5)*Log[x]^4),x]
3.13.30.3.1 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 = 32.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62
| method | result | size |
| parallelrisch | \(2 \,{\mathrm e}^{\frac {50 \left (x^{3}-8 x^{2}+5 x +50\right ) \ln \left (x \right )^{2}}{75 x^{2} \ln \left (x \right )^{2}-750 x \ln \left (x \right )^{2}+1875 \ln \left (x \right )^{2}-x -5}}\) | \(52\) |
int(((7500*x^5-150000*x^4+1125000*x^3-3750000*x^2+4687500*x)*ln(x)^4+(-200 *x^4-700*x^3+8000*x^2+2500*x)*ln(x)^2+(-200*x^4+600*x^3+7000*x^2-15000*x-5 0000)*ln(x))*exp((50*x^3-400*x^2+250*x+2500)*ln(x)^2/((75*x^2-750*x+1875)* ln(x)^2-x-5))/((5625*x^5-112500*x^4+843750*x^3-2812500*x^2+3515625*x)*ln(x )^4+(-150*x^4+750*x^3+3750*x^2-18750*x)*ln(x)^2+x^3+10*x^2+25*x),x,method= _RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {e^{\frac {\left (2500+250 x-400 x^2+50 x^3\right ) \log ^2(x)}{-5-x+\left (1875-750 x+75 x^2\right ) \log ^2(x)}} \left (\left (-50000-15000 x+7000 x^2+600 x^3-200 x^4\right ) \log (x)+\left (2500 x+8000 x^2-700 x^3-200 x^4\right ) \log ^2(x)+\left (4687500 x-3750000 x^2+1125000 x^3-150000 x^4+7500 x^5\right ) \log ^4(x)\right )}{25 x+10 x^2+x^3+\left (-18750 x+3750 x^2+750 x^3-150 x^4\right ) \log ^2(x)+\left (3515625 x-2812500 x^2+843750 x^3-112500 x^4+5625 x^5\right ) \log ^4(x)} \, dx=2 \, e^{\left (\frac {50 \, {\left (x^{3} - 8 \, x^{2} + 5 \, x + 50\right )} \log \left (x\right )^{2}}{75 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )^{2} - x - 5}\right )} \]
integrate(((7500*x^5-150000*x^4+1125000*x^3-3750000*x^2+4687500*x)*log(x)^ 4+(-200*x^4-700*x^3+8000*x^2+2500*x)*log(x)^2+(-200*x^4+600*x^3+7000*x^2-1 5000*x-50000)*log(x))*exp((50*x^3-400*x^2+250*x+2500)*log(x)^2/((75*x^2-75 0*x+1875)*log(x)^2-x-5))/((5625*x^5-112500*x^4+843750*x^3-2812500*x^2+3515 625*x)*log(x)^4+(-150*x^4+750*x^3+3750*x^2-18750*x)*log(x)^2+x^3+10*x^2+25 *x),x, algorithm=\
Time = 0.81 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\frac {\left (2500+250 x-400 x^2+50 x^3\right ) \log ^2(x)}{-5-x+\left (1875-750 x+75 x^2\right ) \log ^2(x)}} \left (\left (-50000-15000 x+7000 x^2+600 x^3-200 x^4\right ) \log (x)+\left (2500 x+8000 x^2-700 x^3-200 x^4\right ) \log ^2(x)+\left (4687500 x-3750000 x^2+1125000 x^3-150000 x^4+7500 x^5\right ) \log ^4(x)\right )}{25 x+10 x^2+x^3+\left (-18750 x+3750 x^2+750 x^3-150 x^4\right ) \log ^2(x)+\left (3515625 x-2812500 x^2+843750 x^3-112500 x^4+5625 x^5\right ) \log ^4(x)} \, dx=2 e^{\frac {\left (50 x^{3} - 400 x^{2} + 250 x + 2500\right ) \log {\left (x \right )}^{2}}{- x + \left (75 x^{2} - 750 x + 1875\right ) \log {\left (x \right )}^{2} - 5}} \]
integrate(((7500*x**5-150000*x**4+1125000*x**3-3750000*x**2+4687500*x)*ln( x)**4+(-200*x**4-700*x**3+8000*x**2+2500*x)*ln(x)**2+(-200*x**4+600*x**3+7 000*x**2-15000*x-50000)*ln(x))*exp((50*x**3-400*x**2+250*x+2500)*ln(x)**2/ ((75*x**2-750*x+1875)*ln(x)**2-x-5))/((5625*x**5-112500*x**4+843750*x**3-2 812500*x**2+3515625*x)*ln(x)**4+(-150*x**4+750*x**3+3750*x**2-18750*x)*ln( x)**2+x**3+10*x**2+25*x),x)
2*exp((50*x**3 - 400*x**2 + 250*x + 2500)*log(x)**2/(-x + (75*x**2 - 750*x + 1875)*log(x)**2 - 5))
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (25) = 50\).
Time = 89.97 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.69 \[ \int \frac {e^{\frac {\left (2500+250 x-400 x^2+50 x^3\right ) \log ^2(x)}{-5-x+\left (1875-750 x+75 x^2\right ) \log ^2(x)}} \left (\left (-50000-15000 x+7000 x^2+600 x^3-200 x^4\right ) \log (x)+\left (2500 x+8000 x^2-700 x^3-200 x^4\right ) \log ^2(x)+\left (4687500 x-3750000 x^2+1125000 x^3-150000 x^4+7500 x^5\right ) \log ^4(x)\right )}{25 x+10 x^2+x^3+\left (-18750 x+3750 x^2+750 x^3-150 x^4\right ) \log ^2(x)+\left (3515625 x-2812500 x^2+843750 x^3-112500 x^4+5625 x^5\right ) \log ^4(x)} \, dx=2 \, e^{\left (\frac {2}{3} \, x + \frac {2 \, x}{225 \, {\left (75 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )^{4} - {\left (x + 5\right )} \log \left (x\right )^{2}\right )}} + \frac {34 \, x}{3 \, {\left (75 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )^{2} - x - 5\right )}} + \frac {2}{45 \, {\left (75 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )^{4} - {\left (x + 5\right )} \log \left (x\right )^{2}\right )}} - \frac {10}{75 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )^{2} - x - 5} + \frac {2}{225 \, \log \left (x\right )^{2}} + \frac {4}{3}\right )} \]
integrate(((7500*x^5-150000*x^4+1125000*x^3-3750000*x^2+4687500*x)*log(x)^ 4+(-200*x^4-700*x^3+8000*x^2+2500*x)*log(x)^2+(-200*x^4+600*x^3+7000*x^2-1 5000*x-50000)*log(x))*exp((50*x^3-400*x^2+250*x+2500)*log(x)^2/((75*x^2-75 0*x+1875)*log(x)^2-x-5))/((5625*x^5-112500*x^4+843750*x^3-2812500*x^2+3515 625*x)*log(x)^4+(-150*x^4+750*x^3+3750*x^2-18750*x)*log(x)^2+x^3+10*x^2+25 *x),x, algorithm=\
2*e^(2/3*x + 2/225*x/(75*(x^2 - 10*x + 25)*log(x)^4 - (x + 5)*log(x)^2) + 34/3*x/(75*(x^2 - 10*x + 25)*log(x)^2 - x - 5) + 2/45/(75*(x^2 - 10*x + 25 )*log(x)^4 - (x + 5)*log(x)^2) - 10/(75*(x^2 - 10*x + 25)*log(x)^2 - x - 5 ) + 2/225/log(x)^2 + 4/3)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (25) = 50\).
Time = 0.58 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.72 \[ \int \frac {e^{\frac {\left (2500+250 x-400 x^2+50 x^3\right ) \log ^2(x)}{-5-x+\left (1875-750 x+75 x^2\right ) \log ^2(x)}} \left (\left (-50000-15000 x+7000 x^2+600 x^3-200 x^4\right ) \log (x)+\left (2500 x+8000 x^2-700 x^3-200 x^4\right ) \log ^2(x)+\left (4687500 x-3750000 x^2+1125000 x^3-150000 x^4+7500 x^5\right ) \log ^4(x)\right )}{25 x+10 x^2+x^3+\left (-18750 x+3750 x^2+750 x^3-150 x^4\right ) \log ^2(x)+\left (3515625 x-2812500 x^2+843750 x^3-112500 x^4+5625 x^5\right ) \log ^4(x)} \, dx=2 \, e^{\left (\frac {50 \, x^{3} \log \left (x\right )^{2}}{75 \, x^{2} \log \left (x\right )^{2} - 750 \, x \log \left (x\right )^{2} + 1875 \, \log \left (x\right )^{2} - x - 5} - \frac {400 \, x^{2} \log \left (x\right )^{2}}{75 \, x^{2} \log \left (x\right )^{2} - 750 \, x \log \left (x\right )^{2} + 1875 \, \log \left (x\right )^{2} - x - 5} + \frac {250 \, x \log \left (x\right )^{2}}{75 \, x^{2} \log \left (x\right )^{2} - 750 \, x \log \left (x\right )^{2} + 1875 \, \log \left (x\right )^{2} - x - 5} + \frac {2500 \, \log \left (x\right )^{2}}{75 \, x^{2} \log \left (x\right )^{2} - 750 \, x \log \left (x\right )^{2} + 1875 \, \log \left (x\right )^{2} - x - 5}\right )} \]
integrate(((7500*x^5-150000*x^4+1125000*x^3-3750000*x^2+4687500*x)*log(x)^ 4+(-200*x^4-700*x^3+8000*x^2+2500*x)*log(x)^2+(-200*x^4+600*x^3+7000*x^2-1 5000*x-50000)*log(x))*exp((50*x^3-400*x^2+250*x+2500)*log(x)^2/((75*x^2-75 0*x+1875)*log(x)^2-x-5))/((5625*x^5-112500*x^4+843750*x^3-2812500*x^2+3515 625*x)*log(x)^4+(-150*x^4+750*x^3+3750*x^2-18750*x)*log(x)^2+x^3+10*x^2+25 *x),x, algorithm=\
2*e^(50*x^3*log(x)^2/(75*x^2*log(x)^2 - 750*x*log(x)^2 + 1875*log(x)^2 - x - 5) - 400*x^2*log(x)^2/(75*x^2*log(x)^2 - 750*x*log(x)^2 + 1875*log(x)^2 - x - 5) + 250*x*log(x)^2/(75*x^2*log(x)^2 - 750*x*log(x)^2 + 1875*log(x) ^2 - x - 5) + 2500*log(x)^2/(75*x^2*log(x)^2 - 750*x*log(x)^2 + 1875*log(x )^2 - x - 5))
Time = 13.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {\left (2500+250 x-400 x^2+50 x^3\right ) \log ^2(x)}{-5-x+\left (1875-750 x+75 x^2\right ) \log ^2(x)}} \left (\left (-50000-15000 x+7000 x^2+600 x^3-200 x^4\right ) \log (x)+\left (2500 x+8000 x^2-700 x^3-200 x^4\right ) \log ^2(x)+\left (4687500 x-3750000 x^2+1125000 x^3-150000 x^4+7500 x^5\right ) \log ^4(x)\right )}{25 x+10 x^2+x^3+\left (-18750 x+3750 x^2+750 x^3-150 x^4\right ) \log ^2(x)+\left (3515625 x-2812500 x^2+843750 x^3-112500 x^4+5625 x^5\right ) \log ^4(x)} \, dx=2\,{\mathrm {e}}^{-\frac {50\,x^3\,{\ln \left (x\right )}^2-400\,x^2\,{\ln \left (x\right )}^2+250\,x\,{\ln \left (x\right )}^2+2500\,{\ln \left (x\right )}^2}{-75\,x^2\,{\ln \left (x\right )}^2+750\,x\,{\ln \left (x\right )}^2+x-1875\,{\ln \left (x\right )}^2+5}} \]
int((exp(-(log(x)^2*(250*x - 400*x^2 + 50*x^3 + 2500))/(x - log(x)^2*(75*x ^2 - 750*x + 1875) + 5))*(log(x)^2*(2500*x + 8000*x^2 - 700*x^3 - 200*x^4) - log(x)*(15000*x - 7000*x^2 - 600*x^3 + 200*x^4 + 50000) + log(x)^4*(468 7500*x - 3750000*x^2 + 1125000*x^3 - 150000*x^4 + 7500*x^5)))/(25*x - log( x)^2*(18750*x - 3750*x^2 - 750*x^3 + 150*x^4) + log(x)^4*(3515625*x - 2812 500*x^2 + 843750*x^3 - 112500*x^4 + 5625*x^5) + 10*x^2 + x^3),x)