Integrand size = 219, antiderivative size = 32 \[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx=e^{-e^{x^2}+x+\frac {x}{10-e^{\frac {e^x}{2 x}}+x}} \] Output:
exp(x/(x-exp(1/2*exp(x)/x)+10)+x-exp(x^2))
Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx=e^{-e^{x^2}+x-\frac {x}{-10+e^{\frac {e^x}{2 x}}-x}} \] Input:
Integrate[(E^((-11*x - x^2 + E^x^2*(10 + x) + E^(E^x/(2*x))*(-E^x^2 + x))/ (-10 + E^(E^x/(2*x)) - x))*(220*x + 40*x^2 + 2*x^3 + E^(E^x/x)*(2*x - 4*E^ x^2*x^2) + E^x^2*(-400*x^2 - 80*x^3 - 4*x^4) + E^(E^x/(2*x))*(E^x*(-1 + x) - 42*x - 4*x^2 + E^x^2*(80*x^2 + 8*x^3))))/(200*x + 2*E^(E^x/x)*x + 40*x^ 2 + 2*x^3 + E^(E^x/(2*x))*(-40*x - 4*x^2)),x]
Output:
E^(-E^x^2 + x - x/(-10 + E^(E^x/(2*x)) - 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.
| \(\displaystyle \int \frac {\left (2 x^3+40 x^2+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{\frac {e^x}{2 x}} \left (-4 x^2+e^{x^2} \left (8 x^3+80 x^2\right )-42 x+e^x (x-1)\right )+e^{x^2} \left (-4 x^4-80 x^3-400 x^2\right )+220 x\right ) \exp \left (\frac {-x^2+e^{x^2} (x+10)+e^{\frac {e^x}{2 x}} \left (x-e^{x^2}\right )-11 x}{-x+e^{\frac {e^x}{2 x}}-10}\right )}{2 x^3+40 x^2+e^{\frac {e^x}{2 x}} \left (-4 x^2-40 x\right )+2 e^{\frac {e^x}{x}} x+200 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (2 x^3+40 x^2+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{\frac {e^x}{2 x}} \left (-4 x^2+e^{x^2} \left (8 x^3+80 x^2\right )-42 x+e^x (x-1)\right )+e^{x^2} \left (-4 x^4-80 x^3-400 x^2\right )+220 x\right ) \exp \left (\frac {-x^2+e^{x^2} (x+10)+e^{\frac {e^x}{2 x}} \left (x-e^{x^2}\right )-11 x}{-x+e^{\frac {e^x}{2 x}}-10}\right )}{2 x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}\right ) \left (2 x^3+40 x^2+220 x+2 e^{\frac {e^x}{x}} \left (x-2 e^{x^2} x^2\right )-4 e^{x^2} \left (x^4+20 x^3+100 x^2\right )-e^{\frac {e^x}{2 x}} \left (4 x^2+42 x+e^x (1-x)-8 e^{x^2} \left (x^3+10 x^2\right )\right )\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {2 \exp \left (\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}\right ) x^2}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}-4 \exp \left (x^2+\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}\right ) x+\frac {40 \exp \left (\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}\right ) x}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}-\frac {4 \exp \left (\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}+\frac {e^x}{2 x}\right ) x}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {220 \exp \left (\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}-\frac {42 \exp \left (\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}+\frac {e^x}{2 x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {\exp \left (x+\frac {x^2+11 x+e^{\frac {e^x}{2 x}} \left (e^{x^2}-x\right )-e^{x^2} (x+10)}{x-e^{\frac {e^x}{2 x}}+10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\exp \left (x+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (x-1)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2 x}-4 \exp \left (x^2+\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) x-\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{2 x}\right ) (2 x+21)}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (x^2+20 x+110\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}+\frac {2 \exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}+\frac {e^x}{x}\right )}{\left (-x+e^{\frac {e^x}{2 x}}-10\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {\exp \left (\frac {e^{\frac {e^x}{2 x}} x-(x+11) x-e^{x^2+\frac {e^x}{2 x}}+e^{x^2} (x+10)}{-x+e^{\frac {e^x}{2 x}}-10}\right ) \left (-4 e^{x^2+\frac {e^x}{x}} x^2-4 e^{x^2} (x+10)^2 x^2+8 e^{x^2+\frac {e^x}{2 x}} (x+10) x^2+2 e^{\frac {e^x}{x}} x-2 e^{\frac {e^x}{2 x}} (2 x+21) x+2 \left (x^2+20 x+110\right ) x+e^{x+\frac {e^x}{2 x}} (x-1)\right )}{x \left (x-e^{\frac {e^x}{2 x}}+10\right )^2}dx\) |
Input:
Int[(E^((-11*x - x^2 + E^x^2*(10 + x) + E^(E^x/(2*x))*(-E^x^2 + x))/(-10 + E^(E^x/(2*x)) - x))*(220*x + 40*x^2 + 2*x^3 + E^(E^x/x)*(2*x - 4*E^x^2*x^ 2) + E^x^2*(-400*x^2 - 80*x^3 - 4*x^4) + E^(E^x/(2*x))*(E^x*(-1 + x) - 42* x - 4*x^2 + E^x^2*(80*x^2 + 8*x^3))))/(200*x + 2*E^(E^x/x)*x + 40*x^2 + 2* x^3 + E^(E^x/(2*x))*(-40*x - 4*x^2)),x]
Output:
$Aborted
Time = 189.57 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (-{\mathrm e}^{x^{2}}+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}+\left (x +10\right ) {\mathrm e}^{x^{2}}-x^{2}-11 x}{{\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}-x -10}}\) | \(52\) |
| risch | \({\mathrm e}^{\frac {{\mathrm e}^{\frac {2 x^{3}+{\mathrm e}^{x}}{2 x}}-{\mathrm e}^{x^{2}} x -{\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}} x +x^{2}-10 \,{\mathrm e}^{x^{2}}+11 x}{x -{\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}+10}}\) | \(63\) |
Input:
int(((-4*x^2*exp(x^2)+2*x)*exp(1/2*exp(x)/x)^2+((8*x^3+80*x^2)*exp(x^2)+(- 1+x)*exp(x)-4*x^2-42*x)*exp(1/2*exp(x)/x)+(-4*x^4-80*x^3-400*x^2)*exp(x^2) +2*x^3+40*x^2+220*x)*exp(((-exp(x^2)+x)*exp(1/2*exp(x)/x)+(x+10)*exp(x^2)- x^2-11*x)/(exp(1/2*exp(x)/x)-x-10))/(2*x*exp(1/2*exp(x)/x)^2+(-4*x^2-40*x) *exp(1/2*exp(x)/x)+2*x^3+40*x^2+200*x),x,method=_RETURNVERBOSE)
Output:
exp(((-exp(x^2)+x)*exp(1/2*exp(x)/x)+(x+10)*exp(x^2)-x^2-11*x)/(exp(1/2*ex p(x)/x)-x-10))
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx=e^{\left (\frac {x^{2} - {\left (x + 10\right )} e^{\left (x^{2}\right )} - {\left (x - e^{\left (x^{2}\right )}\right )} e^{\left (\frac {e^{x}}{2 \, x}\right )} + 11 \, x}{x - e^{\left (\frac {e^{x}}{2 \, x}\right )} + 10}\right )} \] Input:
integrate(((-4*exp(x^2)*x^2+2*x)*exp(1/2*exp(x)/x)^2+((8*x^3+80*x^2)*exp(x ^2)+(-1+x)*exp(x)-4*x^2-42*x)*exp(1/2*exp(x)/x)+(-4*x^4-80*x^3-400*x^2)*ex p(x^2)+2*x^3+40*x^2+220*x)*exp(((-exp(x^2)+x)*exp(1/2*exp(x)/x)+(x+10)*exp (x^2)-x^2-11*x)/(exp(1/2*exp(x)/x)-x-10))/(2*x*exp(1/2*exp(x)/x)^2+(-4*x^2 -40*x)*exp(1/2*exp(x)/x)+2*x^3+40*x^2+200*x),x, algorithm="fricas")
Output:
e^((x^2 - (x + 10)*e^(x^2) - (x - e^(x^2))*e^(1/2*e^x/x) + 11*x)/(x - e^(1 /2*e^x/x) + 10))
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 8.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx=e^{\frac {- x^{2} - 11 x + \left (x + 10\right ) e^{x^{2}} + \left (x - e^{x^{2}}\right ) e^{\frac {e^{x}}{2 x}}}{- x + e^{\frac {e^{x}}{2 x}} - 10}} \] Input:
integrate(((-4*exp(x**2)*x**2+2*x)*exp(1/2*exp(x)/x)**2+((8*x**3+80*x**2)* exp(x**2)+(-1+x)*exp(x)-4*x**2-42*x)*exp(1/2*exp(x)/x)+(-4*x**4-80*x**3-40 0*x**2)*exp(x**2)+2*x**3+40*x**2+220*x)*exp(((-exp(x**2)+x)*exp(1/2*exp(x) /x)+(x+10)*exp(x**2)-x**2-11*x)/(exp(1/2*exp(x)/x)-x-10))/(2*x*exp(1/2*exp (x)/x)**2+(-4*x**2-40*x)*exp(1/2*exp(x)/x)+2*x**3+40*x**2+200*x),x)
Output:
exp((-x**2 - 11*x + (x + 10)*exp(x**2) + (x - exp(x**2))*exp(exp(x)/(2*x)) )/(-x + exp(exp(x)/(2*x)) - 10))
\[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx=\int { \frac {{\left (2 \, x^{3} + 40 \, x^{2} - 4 \, {\left (x^{4} + 20 \, x^{3} + 100 \, x^{2}\right )} e^{\left (x^{2}\right )} - 2 \, {\left (2 \, x^{2} e^{\left (x^{2}\right )} - x\right )} e^{\left (\frac {e^{x}}{x}\right )} - {\left (4 \, x^{2} - 8 \, {\left (x^{3} + 10 \, x^{2}\right )} e^{\left (x^{2}\right )} - {\left (x - 1\right )} e^{x} + 42 \, x\right )} e^{\left (\frac {e^{x}}{2 \, x}\right )} + 220 \, x\right )} e^{\left (\frac {x^{2} - {\left (x + 10\right )} e^{\left (x^{2}\right )} - {\left (x - e^{\left (x^{2}\right )}\right )} e^{\left (\frac {e^{x}}{2 \, x}\right )} + 11 \, x}{x - e^{\left (\frac {e^{x}}{2 \, x}\right )} + 10}\right )}}{2 \, {\left (x^{3} + 20 \, x^{2} + x e^{\left (\frac {e^{x}}{x}\right )} - 2 \, {\left (x^{2} + 10 \, x\right )} e^{\left (\frac {e^{x}}{2 \, x}\right )} + 100 \, x\right )}} \,d x } \] Input:
integrate(((-4*exp(x^2)*x^2+2*x)*exp(1/2*exp(x)/x)^2+((8*x^3+80*x^2)*exp(x ^2)+(-1+x)*exp(x)-4*x^2-42*x)*exp(1/2*exp(x)/x)+(-4*x^4-80*x^3-400*x^2)*ex p(x^2)+2*x^3+40*x^2+220*x)*exp(((-exp(x^2)+x)*exp(1/2*exp(x)/x)+(x+10)*exp (x^2)-x^2-11*x)/(exp(1/2*exp(x)/x)-x-10))/(2*x*exp(1/2*exp(x)/x)^2+(-4*x^2 -40*x)*exp(1/2*exp(x)/x)+2*x^3+40*x^2+200*x),x, algorithm="maxima")
Output:
1/2*integrate((2*x^3 + 40*x^2 - 4*(x^4 + 20*x^3 + 100*x^2)*e^(x^2) - 2*(2* x^2*e^(x^2) - x)*e^(e^x/x) - (4*x^2 - 8*(x^3 + 10*x^2)*e^(x^2) - (x - 1)*e ^x + 42*x)*e^(1/2*e^x/x) + 220*x)*e^((x^2 - (x + 10)*e^(x^2) - (x - e^(x^2 ))*e^(1/2*e^x/x) + 11*x)/(x - e^(1/2*e^x/x) + 10))/(x^3 + 20*x^2 + x*e^(e^ x/x) - 2*(x^2 + 10*x)*e^(1/2*e^x/x) + 100*x), x)
Exception generated. \[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((-4*exp(x^2)*x^2+2*x)*exp(1/2*exp(x)/x)^2+((8*x^3+80*x^2)*exp(x ^2)+(-1+x)*exp(x)-4*x^2-42*x)*exp(1/2*exp(x)/x)+(-4*x^4-80*x^3-400*x^2)*ex p(x^2)+2*x^3+40*x^2+220*x)*exp(((-exp(x^2)+x)*exp(1/2*exp(x)/x)+(x+10)*exp (x^2)-x^2-11*x)/(exp(1/2*exp(x)/x)-x-10))/(2*x*exp(1/2*exp(x)/x)^2+(-4*x^2 -40*x)*exp(1/2*exp(x)/x)+2*x^3+40*x^2+200*x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-32,[0,8,9,6,25]%%%}+%%%{-1696,[0,8,9,6,24]%%%}+%%%{-35232 ,[0,8,9,6
Time = 1.02 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.41 \[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx={\mathrm {e}}^{-\frac {10\,{\mathrm {e}}^{x^2}}{x-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}+10}}\,{\mathrm {e}}^{\frac {11\,x}{x-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}+10}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{x^2}}{x-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}+10}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}}{x-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}+10}}\,{\mathrm {e}}^{\frac {x^2}{x-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}+10}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}}{x-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2\,x}}+10}} \] Input:
int((exp((11*x - exp(exp(x)/(2*x))*(x - exp(x^2)) - exp(x^2)*(x + 10) + x^ 2)/(x - exp(exp(x)/(2*x)) + 10))*(220*x + exp(exp(x)/x)*(2*x - 4*x^2*exp(x ^2)) - exp(x^2)*(400*x^2 + 80*x^3 + 4*x^4) - exp(exp(x)/(2*x))*(42*x - exp (x)*(x - 1) - exp(x^2)*(80*x^2 + 8*x^3) + 4*x^2) + 40*x^2 + 2*x^3))/(200*x - exp(exp(x)/(2*x))*(40*x + 4*x^2) + 2*x*exp(exp(x)/x) + 40*x^2 + 2*x^3), x)
Output:
exp(-(10*exp(x^2))/(x - exp(exp(x)/(2*x)) + 10))*exp((11*x)/(x - exp(exp(x )/(2*x)) + 10))*exp(-(x*exp(x^2))/(x - exp(exp(x)/(2*x)) + 10))*exp((exp(x ^2)*exp(exp(x)/(2*x)))/(x - exp(exp(x)/(2*x)) + 10))*exp(x^2/(x - exp(exp( x)/(2*x)) + 10))*exp(-(x*exp(exp(x)/(2*x)))/(x - exp(exp(x)/(2*x)) + 10))
\[ \int \frac {e^{\frac {-11 x-x^2+e^{x^2} (10+x)+e^{\frac {e^x}{2 x}} \left (-e^{x^2}+x\right )}{-10+e^{\frac {e^x}{2 x}}-x}} \left (220 x+40 x^2+2 x^3+e^{\frac {e^x}{x}} \left (2 x-4 e^{x^2} x^2\right )+e^{x^2} \left (-400 x^2-80 x^3-4 x^4\right )+e^{\frac {e^x}{2 x}} \left (e^x (-1+x)-42 x-4 x^2+e^{x^2} \left (80 x^2+8 x^3\right )\right )\right )}{200 x+2 e^{\frac {e^x}{x}} x+40 x^2+2 x^3+e^{\frac {e^x}{2 x}} \left (-40 x-4 x^2\right )} \, dx=\int \frac {\left (\left (-4 \,{\mathrm e}^{x^{2}} x^{2}+2 x \right ) \left ({\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}\right )^{2}+\left (\left (8 x^{3}+80 x^{2}\right ) {\mathrm e}^{x^{2}}+\left (x -1\right ) {\mathrm e}^{x}-4 x^{2}-42 x \right ) {\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}+\left (-4 x^{4}-80 x^{3}-400 x^{2}\right ) {\mathrm e}^{x^{2}}+2 x^{3}+40 x^{2}+220 x \right ) {\mathrm e}^{\frac {\left (-{\mathrm e}^{x^{2}}+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}+\left (x +10\right ) {\mathrm e}^{x^{2}}-x^{2}-11 x}{{\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}-x -10}}}{2 x \left ({\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}\right )^{2}+\left (-4 x^{2}-40 x \right ) {\mathrm e}^{\frac {{\mathrm e}^{x}}{2 x}}+2 x^{3}+40 x^{2}+200 x}d x \] Input:
int(((-4*exp(x^2)*x^2+2*x)*exp(1/2*exp(x)/x)^2+((8*x^3+80*x^2)*exp(x^2)+(- 1+x)*exp(x)-4*x^2-42*x)*exp(1/2*exp(x)/x)+(-4*x^4-80*x^3-400*x^2)*exp(x^2) +2*x^3+40*x^2+220*x)*exp(((-exp(x^2)+x)*exp(1/2*exp(x)/x)+(x+10)*exp(x^2)- x^2-11*x)/(exp(1/2*exp(x)/x)-x-10))/(2*x*exp(1/2*exp(x)/x)^2+(-4*x^2-40*x) *exp(1/2*exp(x)/x)+2*x^3+40*x^2+200*x),x)
Output:
int(((-4*exp(x^2)*x^2+2*x)*exp(1/2*exp(x)/x)^2+((8*x^3+80*x^2)*exp(x^2)+(- 1+x)*exp(x)-4*x^2-42*x)*exp(1/2*exp(x)/x)+(-4*x^4-80*x^3-400*x^2)*exp(x^2) +2*x^3+40*x^2+220*x)*exp(((-exp(x^2)+x)*exp(1/2*exp(x)/x)+(x+10)*exp(x^2)- x^2-11*x)/(exp(1/2*exp(x)/x)-x-10))/(2*x*exp(1/2*exp(x)/x)^2+(-4*x^2-40*x) *exp(1/2*exp(x)/x)+2*x^3+40*x^2+200*x),x)