Integrand size = 138, antiderivative size = 24 \[ \int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx=-6+e^{\frac {e^{x+\frac {5}{e^x+x+x^2}}}{x}} \]
Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx=e^{\frac {e^{x+\frac {5}{e^x+x+x^2}}}{x}} \]
Integrate[(E^(E^((5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))/x + (5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))*(E^(2*x)*(-1 + x) - 5*x - 11*x^2 - x^3 + x^4 + x^5 + E^x*(-7*x + 2*x^3)))/(E^(2*x)*x^2 + x^4 + 2*x^5 + x^6 + E^x*(2*x^3 + 2*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.
| \(\displaystyle \int \frac {\left (x^5+x^4-x^3+e^x \left (2 x^3-7 x\right )-11 x^2-5 x+e^{2 x} (x-1)\right ) \exp \left (\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}+\frac {e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}}}{x}\right )}{x^6+2 x^5+x^4+e^{2 x} x^2+e^x \left (2 x^4+2 x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (x^5+x^4-x^3+e^x \left (2 x^3-7 x\right )-11 x^2-5 x+e^{2 x} (x-1)\right ) \exp \left (\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}+\frac {e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}}}{x}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}+\frac {e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}}}{x}\right )}{x^2}-\frac {5 \exp \left (\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}+\frac {e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}}}{x}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}+\frac {e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}}}{x}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {\exp \left (\frac {x^2}{x^2+x+e^x}+\frac {e^x x}{x^2+x+e^x}+\frac {5}{x^2+x+e^x}+\frac {x^3}{x^2+x+e^x}\right )}{x}+\frac {x^2}{x^2+x+e^x}+\frac {e^x x}{x^2+x+e^x}+\frac {5}{x^2+x+e^x}+\frac {x^3}{x^2+x+e^x}\right )}{x^2}-\frac {5 \exp \left (\frac {\exp \left (\frac {x^2}{x^2+x+e^x}+\frac {e^x x}{x^2+x+e^x}+\frac {5}{x^2+x+e^x}+\frac {x^3}{x^2+x+e^x}\right )}{x}+\frac {x^2}{x^2+x+e^x}+\frac {e^x x}{x^2+x+e^x}+\frac {5}{x^2+x+e^x}+\frac {x^3}{x^2+x+e^x}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {\exp \left (\frac {x^2}{x^2+x+e^x}+\frac {e^x x}{x^2+x+e^x}+\frac {5}{x^2+x+e^x}+\frac {x^3}{x^2+x+e^x}\right )}{x}+\frac {x^2}{x^2+x+e^x}+\frac {e^x x}{x^2+x+e^x}+\frac {5}{x^2+x+e^x}+\frac {x^3}{x^2+x+e^x}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2}-\frac {5 \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )}+\frac {5 \left (x^2-x-1\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x \left (x^2+x+e^x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (e^x x \left (2 x^2-7\right )+x \left (x^4+x^3-x^2-11 x-5\right )+e^{2 x} (x-1)\right ) \exp \left (\frac {e^x x^2+e^{\frac {x^3+x^2+e^x x+5}{x^2+x+e^x}} (x+1) x+\left (x^3+x^2+5\right ) x+e^{\frac {2 x^3+2 x^2+2 e^x x+5}{x^2+x+e^x}}}{x \left (x^2+x+e^x\right )}\right )}{x^2 \left (x^2+x+e^x\right )^2}dx\) |
Int[(E^(E^((5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))/x + (5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))*(E^(2*x)*(-1 + x) - 5*x - 11*x^2 - x^3 + x^4 + x^5 + E^x*(-7*x + 2*x^3)))/(E^(2*x)*x^2 + x^4 + 2*x^5 + x^6 + E^x*(2*x^3 + 2*x ^4)),x]
3.25.68.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 56.65 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \({\mathrm e}^{\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +x^{3}+x^{2}+5}{{\mathrm e}^{x}+x^{2}+x}}}{x}}\) | \(29\) |
| parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +x^{3}+x^{2}+5}{{\mathrm e}^{x}+x^{2}+x}}}{x}}\) | \(29\) |
int(((-1+x)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp((exp(x )*x+x^3+x^2+5)/(exp(x)+x^2+x))*exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x) )/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x,method=_RETURNVER BOSE)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx=e^{\left (\frac {x^{4} + x^{3} + x^{2} e^{x} + {\left (x^{2} + x + e^{x}\right )} e^{\left (\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}\right )} + 5 \, x}{x^{3} + x^{2} + x e^{x}} - \frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}\right )} \]
integrate(((-1+x)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp( (exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))*exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+ x^2+x))/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x, algorithm= \
e^((x^4 + x^3 + x^2*e^x + (x^2 + x + e^x)*e^((x^3 + x^2 + x*e^x + 5)/(x^2 + x + e^x)) + 5*x)/(x^3 + x^2 + x*e^x) - (x^3 + x^2 + x*e^x + 5)/(x^2 + x + e^x))
Time = 3.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx=e^{\frac {e^{\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}}}{x}} \]
integrate(((-1+x)*exp(x)**2+(2*x**3-7*x)*exp(x)+x**5+x**4-x**3-11*x**2-5*x )*exp((exp(x)*x+x**3+x**2+5)/(exp(x)+x**2+x))*exp(exp((exp(x)*x+x**3+x**2+ 5)/(exp(x)+x**2+x))/x)/(exp(x)**2*x**2+(2*x**4+2*x**3)*exp(x)+x**6+2*x**5+ x**4),x)
Time = 0.66 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx=e^{\left (\frac {e^{\left (x + \frac {5}{x^{2} + x + e^{x}}\right )}}{x}\right )} \]
integrate(((-1+x)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp( (exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))*exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+ x^2+x))/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x, algorithm= \
\[ \int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (x^{5} + x^{4} - x^{3} - 11 \, x^{2} + {\left (x - 1\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{3} - 7 \, x\right )} e^{x} - 5 \, x\right )} e^{\left (\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}} + \frac {e^{\left (\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}\right )}}{x}\right )}}{x^{6} + 2 \, x^{5} + x^{4} + x^{2} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} + x^{3}\right )} e^{x}} \,d x } \]
integrate(((-1+x)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp( (exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))*exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+ x^2+x))/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x, algorithm= \
integrate((x^5 + x^4 - x^3 - 11*x^2 + (x - 1)*e^(2*x) + (2*x^3 - 7*x)*e^x - 5*x)*e^((x^3 + x^2 + x*e^x + 5)/(x^2 + x + e^x) + e^((x^3 + x^2 + x*e^x + 5)/(x^2 + x + e^x))/x)/(x^6 + 2*x^5 + x^4 + x^2*e^(2*x) + 2*(x^4 + x^3)* e^x), x)
Time = 12.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{x+{\mathrm {e}}^x+x^2}}\,{\mathrm {e}}^{\frac {x^2}{x+{\mathrm {e}}^x+x^2}}\,{\mathrm {e}}^{\frac {x^3}{x+{\mathrm {e}}^x+x^2}}\,{\mathrm {e}}^{\frac {5}{x+{\mathrm {e}}^x+x^2}}}{x}} \]
int(-(exp(exp((x*exp(x) + x^2 + x^3 + 5)/(x + exp(x) + x^2))/x)*exp((x*exp (x) + x^2 + x^3 + 5)/(x + exp(x) + x^2))*(5*x - exp(2*x)*(x - 1) + exp(x)* (7*x - 2*x^3) + 11*x^2 + x^3 - x^4 - x^5))/(exp(x)*(2*x^3 + 2*x^4) + x^2*e xp(2*x) + x^4 + 2*x^5 + x^6),x)