Integrand size = 137, antiderivative size = 37 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\frac {3 \left (5-\frac {x}{1+e^{\frac {3+e^x x}{x}}}\right )}{x}}-x}{x} \]
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{-\frac {3}{1+e^{e^x+\frac {3}{x}}}+\frac {15}{x}}}{x} \]
Integrate[(E^((15 + 15*E^((3 + E^x*x)/x) - 3*x)/(x + E^((3 + E^x*x)/x)*x)) *(-15 + E^((2*(3 + E^x*x))/x)*(-15 - x) - x + E^((3 + E^x*x)/x)*(-39 - 2*x + 3*E^x*x^2)))/(x^3 + 2*E^((3 + E^x*x)/x)*x^3 + E^((2*(3 + E^x*x))/x)*x^3 ),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.
| \(\displaystyle \int \frac {\left (e^{\frac {e^x x+3}{x}} \left (3 e^x x^2-2 x-39\right )+e^{\frac {2 \left (e^x x+3\right )}{x}} (-x-15)-x-15\right ) \exp \left (\frac {-3 x+15 e^{\frac {e^x x+3}{x}}+15}{e^{\frac {e^x x+3}{x}} x+x}\right )}{2 e^{\frac {e^x x+3}{x}} x^3+e^{\frac {2 \left (e^x x+3\right )}{x}} x^3+x^3} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (e^{\frac {e^x x+3}{x}} \left (3 e^x x^2-2 x-39\right )+e^{\frac {2 \left (e^x x+3\right )}{x}} (-x-15)-x-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{\frac {e^x x+3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{\frac {e^x x+3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{\frac {e^x x+3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {39 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{\frac {e^x x+3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{\frac {e^x x+3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}-\frac {2 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{\frac {e^x x+3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{\frac {e^x x+3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (3 e^{x+e^x+\frac {3}{x}} x^2-x-e^{2 e^x+\frac {6}{x}} (x+15)-e^{e^x+\frac {3}{x}} (2 x+39)-15\right ) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(-x-15) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+\frac {2 \left (e^x x+3\right )}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {(2 x+39) \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {15 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x^2}+\frac {3 \exp \left (\frac {3 \left (-x+5 e^{e^x+\frac {3}{x}}+5\right )}{e^{e^x+\frac {3}{x}} x+x}+e^x+x+\frac {3}{x}\right )}{\left (e^{e^x+\frac {3}{x}}+1\right )^2 x}\right )dx\) |
Int[(E^((15 + 15*E^((3 + E^x*x)/x) - 3*x)/(x + E^((3 + E^x*x)/x)*x))*(-15 + E^((2*(3 + E^x*x))/x)*(-15 - x) - x + E^((3 + E^x*x)/x)*(-39 - 2*x + 3*E ^x*x^2)))/(x^3 + 2*E^((3 + E^x*x)/x)*x^3 + E^((2*(3 + E^x*x))/x)*x^3),x]
3.4.49.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 3.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(\frac {{\mathrm e}^{-\frac {3 \left (-5 \,{\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}-5+x \right )}{x \left ({\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}+1\right )}}}{x}\) | \(42\) |
| parallelrisch | \(\frac {{\mathrm e}^{\frac {15 \,{\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}+15-3 x}{x \left ({\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}+1\right )}}}{x}\) | \(44\) |
int(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x)*x+3)/ x)-x-15)*exp((15*exp((exp(x)*x+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+x))/(x ^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x,method=_RETURNVE RBOSE)
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\left (-\frac {3 \, {\left (x - 5 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} - 5\right )}}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x}\right )}}{x} \]
integrate(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x) *x+3)/x)-x-15)*exp((15*exp((exp(x)*x+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+ x))/(x^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x, algorithm =\
Time = 0.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\frac {- 3 x + 15 e^{\frac {x e^{x} + 3}{x}} + 15}{x e^{\frac {x e^{x} + 3}{x}} + x}}}{x} \]
integrate(((-x-15)*exp((exp(x)*x+3)/x)**2+(3*exp(x)*x**2-2*x-39)*exp((exp( x)*x+3)/x)-x-15)*exp((15*exp((exp(x)*x+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x )+x))/(x**3*exp((exp(x)*x+3)/x)**2+2*x**3*exp((exp(x)*x+3)/x)+x**3),x)
\[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\int { -\frac {{\left ({\left (x + 15\right )} e^{\left (\frac {2 \, {\left (x e^{x} + 3\right )}}{x}\right )} - {\left (3 \, x^{2} e^{x} - 2 \, x - 39\right )} e^{\left (\frac {x e^{x} + 3}{x}\right )} + x + 15\right )} e^{\left (-\frac {3 \, {\left (x - 5 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} - 5\right )}}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x}\right )}}{x^{3} e^{\left (\frac {2 \, {\left (x e^{x} + 3\right )}}{x}\right )} + 2 \, x^{3} e^{\left (\frac {x e^{x} + 3}{x}\right )} + x^{3}} \,d x } \]
integrate(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x) *x+3)/x)-x-15)*exp((15*exp((exp(x)*x+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+ x))/(x^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x, algorithm =\
-integrate(((x + 15)*e^(2*(x*e^x + 3)/x) - (3*x^2*e^x - 2*x - 39)*e^((x*e^ x + 3)/x) + x + 15)*e^(-3*(x - 5*e^((x*e^x + 3)/x) - 5)/(x*e^((x*e^x + 3)/ x) + x))/(x^3*e^(2*(x*e^x + 3)/x) + 2*x^3*e^((x*e^x + 3)/x) + x^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (34) = 68\).
Time = 0.61 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.95 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {e^{\left (\frac {x e^{\left (x + \frac {x e^{x} + 3}{x}\right )} + x e^{x} - 3 \, x + 18 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} + 18}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x} - \frac {x e^{x} + 3}{x}\right )}}{x} \]
integrate(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x) *x+3)/x)-x-15)*exp((15*exp((exp(x)*x+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+ x))/(x^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x, algorithm =\
e^((x*e^(x + (x*e^x + 3)/x) + x*e^x - 3*x + 18*e^((x*e^x + 3)/x) + 18)/(x* e^((x*e^x + 3)/x) + x) - (x*e^x + 3)/x)/x
Time = 9.79 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.78 \[ \int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx=\frac {{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}{x+x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}}\,{\mathrm {e}}^{\frac {15}{x+x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}}\,{\mathrm {e}}^{-\frac {3}{{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}+1}}}{x} \]
int(-(exp((15*exp((x*exp(x) + 3)/x) - 3*x + 15)/(x + x*exp((x*exp(x) + 3)/ x)))*(x + exp((x*exp(x) + 3)/x)*(2*x - 3*x^2*exp(x) + 39) + exp((2*(x*exp( x) + 3))/x)*(x + 15) + 15))/(x^3 + 2*x^3*exp((x*exp(x) + 3)/x) + x^3*exp(( 2*(x*exp(x) + 3))/x)),x)