Integrand size = 179, antiderivative size = 33 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4+4 e^{-\frac {-\frac {2}{5}+x}{\frac {12}{5+e^x}+x (-2+2 x)}} x \]
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 e^{-\frac {\left (5+e^x\right ) (-2+5 x)}{10 \left (6-\left (5+e^x\right ) x+\left (5+e^x\right ) x^2\right )}} x \]
Integrate[(720 - 1400*x + 1500*x^2 - 750*x^3 + 500*x^4 + E^(2*x)*(4*x + 12 *x^2 - 30*x^3 + 20*x^4) + E^x*(-236*x + 300*x^2 - 300*x^3 + 200*x^4))/(E^( (-10 + 25*x + E^x*(-2 + 5*x))/(60 - 50*x + 50*x^2 + E^x*(-10*x + 10*x^2))) *(180 - 300*x + 425*x^2 - 250*x^3 + 125*x^4 + E^(2*x)*(5*x^2 - 10*x^3 + 5* x^4) + E^x*(-60*x + 110*x^2 - 100*x^3 + 50*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.
| \(\displaystyle \int \frac {\left (500 x^4-750 x^3+1500 x^2+e^{2 x} \left (20 x^4-30 x^3+12 x^2+4 x\right )+e^x \left (200 x^4-300 x^3+300 x^2-236 x\right )-1400 x+720\right ) \exp \left (-\frac {25 x+e^x (5 x-2)-10}{50 x^2+e^x \left (10 x^2-10 x\right )-50 x+60}\right )}{125 x^4-250 x^3+425 x^2+e^{2 x} \left (5 x^4-10 x^3+5 x^2\right )+e^x \left (50 x^4-100 x^3+110 x^2-60 x\right )-300 x+180} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (500 x^4-750 x^3+1500 x^2+e^{2 x} \left (20 x^4-30 x^3+12 x^2+4 x\right )+e^x \left (200 x^4-300 x^3+300 x^2-236 x\right )-1400 x+720\right ) \exp \left (-\frac {\left (e^x+5\right ) (5 x-2)}{10 \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}\right )}{5 \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {2 \exp \left (\frac {\left (5+e^x\right ) (2-5 x)}{10 \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}\right ) \left (250 x^4-375 x^3+750 x^2-700 x-2 e^x \left (-50 x^4+75 x^3-75 x^2+59 x\right )+e^{2 x} \left (10 x^4-15 x^3+6 x^2+2 x\right )+360\right )}{\left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (\frac {\left (5+e^x\right ) (2-5 x)}{10 \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}\right ) \left (250 x^4-375 x^3+750 x^2-700 x-2 e^x \left (-50 x^4+75 x^3-75 x^2+59 x\right )+e^{2 x} \left (10 x^4-15 x^3+6 x^2+2 x\right )+360\right )}{\left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (\frac {\left (5+e^x\right ) (2-5 x)}{10 \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (\frac {\left (5+e^x\right ) (2-5 x)}{10 \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (\frac {\left (5+e^x\right ) (2-5 x)}{10 \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{5} \int \frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 \left (5+e^x\right )^2 x^4-15 \left (5+e^x\right )^2 x^3+6 \left (125+25 e^x+e^{2 x}\right ) x^2+2 \left (-350-59 e^x+e^{2 x}\right ) x+360\right )}{\left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (5 x^3+8 x^2-11 x+4\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )}+\frac {\exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (10 x^3-15 x^2+6 x+2\right )}{(x-1)^2 x}+\frac {6 \exp \left (-\frac {\left (5+e^x\right ) (5 x-2)}{10 \left (\left (5+e^x\right ) x^2-\left (5+e^x\right ) x+6\right )}\right ) \left (25 x^5-60 x^4+75 x^3+8 x^2-42 x+12\right )}{(x-1)^2 x \left (e^x x^2+5 x^2-e^x x-5 x+6\right )^2}\right )dx\) |
Int[(720 - 1400*x + 1500*x^2 - 750*x^3 + 500*x^4 + E^(2*x)*(4*x + 12*x^2 - 30*x^3 + 20*x^4) + E^x*(-236*x + 300*x^2 - 300*x^3 + 200*x^4))/(E^((-10 + 25*x + E^x*(-2 + 5*x))/(60 - 50*x + 50*x^2 + E^x*(-10*x + 10*x^2)))*(180 - 300*x + 425*x^2 - 250*x^3 + 125*x^4 + E^(2*x)*(5*x^2 - 10*x^3 + 5*x^4) + E^x*(-60*x + 110*x^2 - 100*x^3 + 50*x^4))),x]
3.1.1.3.1 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 = 2.71 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(4 x \,{\mathrm e}^{-\frac {\left (5 x -2\right ) \left ({\mathrm e}^{x}+5\right )}{10 \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6\right )}}\) | \(39\) |
| norman | \(\frac {\left (24 x -20 x^{2}+20 x^{3}-4 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{x} x^{3}\right ) {\mathrm e}^{-\frac {\left (5 x -2\right ) {\mathrm e}^{x}+25 x -10}{\left (10 x^{2}-10 x \right ) {\mathrm e}^{x}+50 x^{2}-50 x +60}}}{{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6}\) | \(94\) |
| parallelrisch | \(\frac {\left (10000 \,{\mathrm e}^{x} x^{3}+50000 x^{3}-10000 \,{\mathrm e}^{x} x^{2}-50000 x^{2}+60000 x \right ) {\mathrm e}^{-\frac {5 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x}+25 x -10}{10 \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +5 x^{2}-5 x +6\right )}}}{2500 \,{\mathrm e}^{x} x^{2}-2500 \,{\mathrm e}^{x} x +12500 x^{2}-12500 x +15000}\) | \(96\) |
int(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-236*x)*e xp(x)+500*x^4-750*x^3+1500*x^2-1400*x+720)/((5*x^4-10*x^3+5*x^2)*exp(x)^2+ (50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x+180)/ex p(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x,method =_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 \, x e^{\left (-\frac {{\left (5 \, x - 2\right )} e^{x} + 25 \, x - 10}{10 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}}\right )} \]
integrate(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-23 6*x)*exp(x)+500*x^4-750*x^3+1500*x^2-1400*x+720)/((5*x^4-10*x^3+5*x^2)*exp (x)^2+(50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x+1 80)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x, algorithm=\
Time = 11.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 x e^{- \frac {25 x + \left (5 x - 2\right ) e^{x} - 10}{50 x^{2} - 50 x + \left (10 x^{2} - 10 x\right ) e^{x} + 60}} \]
integrate(((20*x**4-30*x**3+12*x**2+4*x)*exp(x)**2+(200*x**4-300*x**3+300* x**2-236*x)*exp(x)+500*x**4-750*x**3+1500*x**2-1400*x+720)/((5*x**4-10*x** 3+5*x**2)*exp(x)**2+(50*x**4-100*x**3+110*x**2-60*x)*exp(x)+125*x**4-250*x **3+425*x**2-300*x+180)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x**2-10*x)*exp(x )+50*x**2-50*x+60)),x)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (29) = 58\).
Time = 0.46 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.18 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 \, x e^{\left (-\frac {x e^{x}}{2 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} - \frac {5 \, x}{2 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} + \frac {e^{x}}{5 \, {\left (5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6\right )}} + \frac {1}{5 \, x^{2} + {\left (x^{2} - x\right )} e^{x} - 5 \, x + 6}\right )} \]
integrate(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-23 6*x)*exp(x)+500*x^4-750*x^3+1500*x^2-1400*x+720)/((5*x^4-10*x^3+5*x^2)*exp (x)^2+(50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x+1 80)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x, algorithm=\
4*x*e^(-1/2*x*e^x/(5*x^2 + (x^2 - x)*e^x - 5*x + 6) - 5/2*x/(5*x^2 + (x^2 - x)*e^x - 5*x + 6) + 1/5*e^x/(5*x^2 + (x^2 - x)*e^x - 5*x + 6) + 1/(5*x^2 + (x^2 - x)*e^x - 5*x + 6))
Time = 1.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4 \, x e^{\left (-\frac {5 \, x^{2} e^{x} + 25 \, x^{2} + 10 \, x e^{x} + 50 \, x - 6 \, e^{x}}{30 \, {\left (x^{2} e^{x} + 5 \, x^{2} - x e^{x} - 5 \, x + 6\right )}} + \frac {1}{6}\right )} \]
integrate(((20*x^4-30*x^3+12*x^2+4*x)*exp(x)^2+(200*x^4-300*x^3+300*x^2-23 6*x)*exp(x)+500*x^4-750*x^3+1500*x^2-1400*x+720)/((5*x^4-10*x^3+5*x^2)*exp (x)^2+(50*x^4-100*x^3+110*x^2-60*x)*exp(x)+125*x^4-250*x^3+425*x^2-300*x+1 80)/exp(((5*x-2)*exp(x)+25*x-10)/((10*x^2-10*x)*exp(x)+50*x^2-50*x+60)),x, algorithm=\
4*x*e^(-1/30*(5*x^2*e^x + 25*x^2 + 10*x*e^x + 50*x - 6*e^x)/(x^2*e^x + 5*x ^2 - x*e^x - 5*x + 6) + 1/6)
Time = 0.71 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.42 \[ \int \frac {e^{-\frac {-10+25 x+e^x (-2+5 x)}{60-50 x+50 x^2+e^x \left (-10 x+10 x^2\right )}} \left (720-1400 x+1500 x^2-750 x^3+500 x^4+e^{2 x} \left (4 x+12 x^2-30 x^3+20 x^4\right )+e^x \left (-236 x+300 x^2-300 x^3+200 x^4\right )\right )}{180-300 x+425 x^2-250 x^3+125 x^4+e^{2 x} \left (5 x^2-10 x^3+5 x^4\right )+e^x \left (-60 x+110 x^2-100 x^3+50 x^4\right )} \, dx=4\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{5\,x^2\,{\mathrm {e}}^x-25\,x-5\,x\,{\mathrm {e}}^x+25\,x^2+30}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^x}{2\,x^2\,{\mathrm {e}}^x-10\,x-2\,x\,{\mathrm {e}}^x+10\,x^2+12}}\,{\mathrm {e}}^{-\frac {5\,x}{2\,x^2\,{\mathrm {e}}^x-10\,x-2\,x\,{\mathrm {e}}^x+10\,x^2+12}}\,{\mathrm {e}}^{\frac {1}{x^2\,{\mathrm {e}}^x-5\,x-x\,{\mathrm {e}}^x+5\,x^2+6}} \]
int((exp((25*x + exp(x)*(5*x - 2) - 10)/(50*x + exp(x)*(10*x - 10*x^2) - 5 0*x^2 - 60))*(exp(2*x)*(4*x + 12*x^2 - 30*x^3 + 20*x^4) - exp(x)*(236*x - 300*x^2 + 300*x^3 - 200*x^4) - 1400*x + 1500*x^2 - 750*x^3 + 500*x^4 + 720 ))/(exp(2*x)*(5*x^2 - 10*x^3 + 5*x^4) - exp(x)*(60*x - 110*x^2 + 100*x^3 - 50*x^4) - 300*x + 425*x^2 - 250*x^3 + 125*x^4 + 180),x)