Integrand size = 158, antiderivative size = 34 \[ \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx=e^{\frac {4}{3} \left (-2+e^{x/2}-x+x^2\right ) \left (-3+\frac {3}{-e^x+x}\right )} \]
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx=e^{-\frac {4 \left (1+e^x-x\right ) \left (-2+e^{x/2}-x+x^2\right )}{e^x-x}} \]
Integrate[(E^((8 - 4*x - 8*x^2 + 4*x^3 + E^(x/2)*(-4 + 4*x) + E^x*(8 - 4*E ^(x/2) + 4*x - 4*x^2))/(E^x - x))*(8 + E^(2*x)*(4 - 2*E^(x/2) - 8*x) + 8*x ^2 - 8*x^3 + E^(x/2)*(-4 + 2*x - 2*x^2) + E^x*(-4 - 20*x + 20*x^2 + E^(x/2 )*(2 + 4*x))))/(E^(2*x) - 2*E^x*x + x^2),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.
| \(\displaystyle \int \frac {\left (-8 x^3+8 x^2+e^{x/2} \left (-2 x^2+2 x-4\right )+e^x \left (20 x^2-20 x+e^{x/2} (4 x+2)-4\right )+e^{2 x} \left (-8 x-2 e^{x/2}+4\right )+8\right ) \exp \left (\frac {4 x^3-8 x^2+e^x \left (-4 x^2+4 x-4 e^{x/2}+8\right )-4 x+e^{x/2} (4 x-4)+8}{e^x-x}\right )}{x^2-2 e^x x+e^{2 x}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-8 x^3+8 x^2+e^{x/2} \left (-2 x^2+2 x-4\right )+e^x \left (20 x^2-20 x+e^{x/2} (4 x+2)-4\right )+e^{2 x} \left (-8 x-2 e^{x/2}+4\right )+8\right ) \exp \left (\frac {4 x^3-8 x^2+e^x \left (-4 x^2+4 x-4 e^{x/2}+8\right )-4 x+e^{x/2} (4 x-4)+8}{e^x-x}\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-4 (2 x-1) \exp \left (\frac {4 x^3-8 x^2+e^x \left (-4 x^2+4 x-4 e^{x/2}+8\right )-4 x+e^{x/2} (4 x-4)+8}{e^x-x}\right )-2 \exp \left (\frac {4 x^3-8 x^2+e^x \left (-4 x^2+4 x-4 e^{x/2}+8\right )-4 x+e^{x/2} (4 x-4)+8}{e^x-x}+\frac {x}{2}\right )+\frac {4 (x-1) \left (x^2-x+e^{x/2}-2\right ) \exp \left (\frac {4 x^3-8 x^2+e^x \left (-4 x^2+4 x-4 e^{x/2}+8\right )-4 x+e^{x/2} (4 x-4)+8}{e^x-x}\right )}{\left (e^x-x\right )^2}+\frac {2 \left (2 x^2-6 x+e^{x/2}-2\right ) \exp \left (\frac {4 x^3-8 x^2+e^x \left (-4 x^2+4 x-4 e^{x/2}+8\right )-4 x+e^{x/2} (4 x-4)+8}{e^x-x}\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-e^x \left (-10 x^2+10 x+2\right )-e^{x/2} \left (x^2-x+2\right )-4 \left (x^3-x^2-1\right )+e^{3 x/2} (2 x+1)-e^{5 x/2}-e^{2 x} (4 x-2)\right ) \exp \left (-\frac {4 \left (-x^3+2 x^2+e^x \left (x^2-x-2\right )+x+e^{3 x/2}-e^{x/2} (x-1)-2\right )}{e^x-x}\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\exp \left (\frac {4 \left (x^3-2 x^2-x-e^{3 x/2}-e^{x/2} (1-x)+e^x \left (-x^2+x+2\right )+2\right )}{e^x-x}\right ) \left (-2 e^{2 x} (1-2 x)+e^{5 x/2}-e^{3 x/2} (2 x+1)+2 e^x \left (-5 x^2+5 x+1\right )+e^{x/2} \left (x^2-x+2\right )-4 \left (-x^3+x^2+1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\exp \left (\frac {4 \left (x^3-2 x^2-x-e^{3 x/2}-e^{x/2} (1-x)+e^x \left (-x^2+x+2\right )+2\right )}{e^x-x}\right ) \left (-2 e^{2 x} (1-2 x)+e^{5 x/2}-e^{3 x/2} (2 x+1)+2 e^x \left (-5 x^2+5 x+1\right )+e^{x/2} \left (x^2-x+2\right )-4 \left (-x^3+x^2+1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (\frac {4 \left (x^3-2 x^2-x-e^{3 x/2}-e^{x/2} (1-x)+e^x \left (-x^2+x+2\right )+2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}+\frac {4 \left (x^3-2 x^2-x-e^{3 x/2}-e^{x/2} (1-x)+e^x \left (-x^2+x+2\right )+2\right )}{e^x-x}\right )-\frac {2 \exp \left (\frac {4 \left (x^3-2 x^2-x-e^{3 x/2}-e^{x/2} (1-x)+e^x \left (-x^2+x+2\right )+2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (\frac {4 \left (x^3-2 x^2-x-e^{3 x/2}-e^{x/2} (1-x)+e^x \left (-x^2+x+2\right )+2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (-e^{3 x/2} (2 x+1)+e^{5 x/2}+e^{2 x} (4 x-2)+e^x \left (-10 x^2+10 x+2\right )+e^{x/2} \left (x^2-x+2\right )+4 \left (x^3-x^2-1\right )\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (2 x-1)+\exp \left (\frac {x}{2}-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right )-\frac {2 \exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) (x-1) \left (x^2-x+e^{x/2}-2\right )}{\left (e^x-x\right )^2}-\frac {\exp \left (-\frac {4 \left (-x^3+2 x^2+x+e^{3 x/2}-e^{x/2} (x-1)+e^x \left (x^2-x-2\right )-2\right )}{e^x-x}\right ) \left (2 x^2-6 x+e^{x/2}-2\right )}{e^x-x}\right )dx\) |
Int[(E^((8 - 4*x - 8*x^2 + 4*x^3 + E^(x/2)*(-4 + 4*x) + E^x*(8 - 4*E^(x/2) + 4*x - 4*x^2))/(E^x - x))*(8 + E^(2*x)*(4 - 2*E^(x/2) - 8*x) + 8*x^2 - 8 *x^3 + E^(x/2)*(-4 + 2*x - 2*x^2) + E^x*(-4 - 20*x + 20*x^2 + E^(x/2)*(2 + 4*x))))/(E^(2*x) - 2*E^x*x + x^2),x]
3.13.63.3.1 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 = 3.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \({\mathrm e}^{-\frac {4 \left (x^{2}+{\mathrm e}^{\frac {x}{2}}-x -2\right ) \left (1+{\mathrm e}^{x}-x \right )}{{\mathrm e}^{x}-x}}\) | \(31\) |
| parallelrisch | \({\mathrm e}^{\frac {\left (-4 \,{\mathrm e}^{\frac {x}{2}}-4 x^{2}+4 x +8\right ) {\mathrm e}^{x}+\left (-4+4 x \right ) {\mathrm e}^{\frac {x}{2}}+4 x^{3}-8 x^{2}-4 x +8}{{\mathrm e}^{x}-x}}\) | \(59\) |
int(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-20*x-4) *exp(x)+(-2*x^2+2*x-4)*exp(1/4*x)^2-8*x^3+8*x^2+8)*exp(((-4*exp(1/4*x)^2-4 *x^2+4*x+8)*exp(x)+(-4+4*x)*exp(1/4*x)^2+4*x^3-8*x^2-4*x+8)/(exp(x)-x))/(e xp(x)^2-2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx=e^{\left (-\frac {4 \, {\left (x^{3} - 2 \, x^{2} + {\left (x - 1\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (x^{2} - x - 2\right )} e^{x} - x - e^{\left (\frac {3}{2} \, x\right )} + 2\right )}}{x - e^{x}}\right )} \]
integrate(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-2 0*x-4)*exp(x)+(-2*x^2+2*x-4)*exp(1/4*x)^2-8*x^3+8*x^2+8)*exp(((-4*exp(1/4* x)^2-4*x^2+4*x+8)*exp(x)+(-4+4*x)*exp(1/4*x)^2+4*x^3-8*x^2-4*x+8)/(exp(x)- x))/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm=\
Time = 0.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx=e^{\frac {4 x^{3} - 8 x^{2} - 4 x + \left (4 x - 4\right ) e^{\frac {x}{2}} + \left (- 4 x^{2} + 4 x - 4 e^{\frac {x}{2}} + 8\right ) e^{x} + 8}{- x + e^{x}}} \]
integrate(((-2*exp(1/4*x)**2-8*x+4)*exp(x)**2+((4*x+2)*exp(1/4*x)**2+20*x* *2-20*x-4)*exp(x)+(-2*x**2+2*x-4)*exp(1/4*x)**2-8*x**3+8*x**2+8)*exp(((-4* exp(1/4*x)**2-4*x**2+4*x+8)*exp(x)+(-4+4*x)*exp(1/4*x)**2+4*x**3-8*x**2-4* x+8)/(exp(x)-x))/(exp(x)**2-2*exp(x)*x+x**2),x)
exp((4*x**3 - 8*x**2 - 4*x + (4*x - 4)*exp(x/2) + (-4*x**2 + 4*x - 4*exp(x /2) + 8)*exp(x) + 8)/(-x + exp(x)))
\[ \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx=\int { -\frac {2 \, {\left (4 \, x^{3} - 4 \, x^{2} + {\left (4 \, x + e^{\left (\frac {1}{2} \, x\right )} - 2\right )} e^{\left (2 \, x\right )} + {\left (x^{2} - x + 2\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (10 \, x^{2} + {\left (2 \, x + 1\right )} e^{\left (\frac {1}{2} \, x\right )} - 10 \, x - 2\right )} e^{x} - 4\right )} e^{\left (-\frac {4 \, {\left (x^{3} - 2 \, x^{2} + {\left (x - 1\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (x^{2} - x + e^{\left (\frac {1}{2} \, x\right )} - 2\right )} e^{x} - x + 2\right )}}{x - e^{x}}\right )}}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}} \,d x } \]
integrate(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-2 0*x-4)*exp(x)+(-2*x^2+2*x-4)*exp(1/4*x)^2-8*x^3+8*x^2+8)*exp(((-4*exp(1/4* x)^2-4*x^2+4*x+8)*exp(x)+(-4+4*x)*exp(1/4*x)^2+4*x^3-8*x^2-4*x+8)/(exp(x)- x))/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm=\
-2*integrate((4*x^3 - 4*x^2 + (4*x + e^(1/2*x) - 2)*e^(2*x) + (x^2 - x + 2 )*e^(1/2*x) - (10*x^2 + (2*x + 1)*e^(1/2*x) - 10*x - 2)*e^x - 4)*e^(-4*(x^ 3 - 2*x^2 + (x - 1)*e^(1/2*x) - (x^2 - x + e^(1/2*x) - 2)*e^x - x + 2)/(x - e^x))/(x^2 - 2*x*e^x + e^(2*x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
Time = 1.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx=e^{\left (-\frac {4 \, {\left (x^{3} - x^{2} e^{x} - 2 \, x^{2} + x e^{\left (\frac {1}{2} \, x\right )} + x e^{x} - x - e^{\left (\frac {3}{2} \, x\right )} - e^{\left (\frac {1}{2} \, x\right )} + 2 \, e^{x} + 2\right )}}{x - e^{x}}\right )} \]
integrate(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-2 0*x-4)*exp(x)+(-2*x^2+2*x-4)*exp(1/4*x)^2-8*x^3+8*x^2+8)*exp(((-4*exp(1/4* x)^2-4*x^2+4*x+8)*exp(x)+(-4+4*x)*exp(1/4*x)^2+4*x^3-8*x^2-4*x+8)/(exp(x)- x))/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm=\
e^(-4*(x^3 - x^2*e^x - 2*x^2 + x*e^(1/2*x) + x*e^x - x - e^(3/2*x) - e^(1/ 2*x) + 2*e^x + 2)/(x - e^x))
Time = 12.43 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.21 \[ \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx={\mathrm {e}}^{-\frac {4\,x^3}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {8\,x^2}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {8}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{x/2}\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{x/2}}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {4\,x\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {4\,x\,{\mathrm {e}}^{x/2}}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}} \]
int(-(exp(-(exp(x)*(4*x - 4*exp(x/2) - 4*x^2 + 8) - 4*x + exp(x/2)*(4*x - 4) - 8*x^2 + 4*x^3 + 8)/(x - exp(x)))*(exp(x/2)*(2*x^2 - 2*x + 4) + exp(2* x)*(8*x + 2*exp(x/2) - 4) + exp(x)*(20*x - exp(x/2)*(4*x + 2) - 20*x^2 + 4 ) - 8*x^2 + 8*x^3 - 8))/(exp(2*x) - 2*x*exp(x) + x^2),x)
exp(-(4*x^3)/(x - exp(x)))*exp((8*x^2)/(x - exp(x)))*exp(-(8*exp(x))/(x - exp(x)))*exp(-8/(x - exp(x)))*exp((4*exp(x/2)*exp(x))/(x - exp(x)))*exp((4 *exp(x/2))/(x - exp(x)))*exp(-(4*x*exp(x))/(x - exp(x)))*exp((4*x)/(x - ex p(x)))*exp(-(4*x*exp(x/2))/(x - exp(x)))*exp((4*x^2*exp(x))/(x - exp(x)))