Integrand size = 117, antiderivative size = 34 \[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx=e^{4+e^{\frac {-\frac {x}{4}+\left (-2+e^x-x\right ) \left (3+\frac {1+x}{x^2}\right )}{x}}} \] Output:
exp(4)*exp(exp(((exp(x)-2-x)*((1+x)/x^2+3)-1/4*x)/x))
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18 \[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx=e^{4+e^{-\frac {13}{4}-\frac {2}{x^3}-\frac {3}{x^2}-\frac {7}{x}+\frac {e^x \left (1+x+3 x^2\right )}{x^3}}} \] Input:
Integrate[(E^(E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4 *x^3)) + (-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))*( E^4*(6 + 6*x + 7*x^2) + E^(4 + x)*(-3 - x - 2*x^2 + 3*x^3)))/x^4,x]
Output:
E^(4 + E^(-13/4 - 2/x^3 - 3/x^2 - 7/x + (E^x*(1 + x + 3*x^2))/x^3))
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.
| \(\displaystyle \int \frac {\left (e^4 \left (7 x^2+6 x+6\right )+e^{x+4} \left (3 x^3-2 x^2-x-3\right )\right ) \exp \left (\exp \left (\frac {-13 x^3-28 x^2+e^x \left (12 x^2+4 x+4\right )-12 x-8}{4 x^3}\right )+\frac {-13 x^3-28 x^2+e^x \left (12 x^2+4 x+4\right )-12 x-8}{4 x^3}\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (\exp \left (\frac {-13 x^3-28 x^2+e^x \left (12 x^2+4 x+4\right )-12 x-8}{4 x^3}\right )+\frac {-13 x^3-28 x^2+e^x \left (12 x^2+4 x+4\right )-12 x-8}{4 x^3}+4\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (\exp \left (\frac {-13 x^3-28 x^2+e^x \left (12 x^2+4 x+4\right )-12 x-8}{4 x^3}\right )+\frac {-13 x^3-28 x^2+e^x \left (12 x^2+4 x+4\right )-12 x-8}{4 x^3}+x+4\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (7 x^2+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}+\frac {\left (3 x^3-2 x^2-x-3\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+x-\frac {7}{x}+\frac {3}{4}\right )}{x^4}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (7 x^2+e^x \left (3 x^3-2 x^2-x-3\right )+6 x+6\right ) \exp \left (-\frac {2}{x^3}-\frac {3}{x^2}+\frac {e^x \left (3 x^2+x+1\right )}{x^3}+e^{\frac {e^x-2}{x^3}+\frac {e^x-3}{x^2}+\frac {3 e^x-7}{x}-\frac {13}{4}}-\frac {7}{x}+\frac {3}{4}\right )}{x^4}dx\) |
Input:
Int[(E^(E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3)) + (-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))*(E^4*(6 + 6*x + 7*x^2) + E^(4 + x)*(-3 - x - 2*x^2 + 3*x^3)))/x^4,x]
Output:
$Aborted
Time = 39.44 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15
| method | result | size |
| parallelrisch | \({\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{\frac {\left (12 x^{2}+4 x +4\right ) {\mathrm e}^{x}-13 x^{3}-28 x^{2}-12 x -8}{4 x^{3}}}}\) | \(39\) |
| risch | \({\mathrm e}^{4+{\mathrm e}^{\frac {12 \,{\mathrm e}^{x} x^{2}-13 x^{3}+4 \,{\mathrm e}^{x} x -28 x^{2}+4 \,{\mathrm e}^{x}-12 x -8}{4 x^{3}}}}\) | \(41\) |
Input:
int(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*((12*x^ 2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+4)*exp (x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x,method=_RETURNVERBOSE)
Output:
exp(4)*exp(exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (29) = 58\).
Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.88 \[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx=e^{\left (\frac {{\left (4 \, x^{3} e^{\left (-\frac {{\left ({\left (13 \, x^{3} + 28 \, x^{2} + 12 \, x + 8\right )} e^{4} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{4 \, x^{3}} + 4\right )} - {\left (13 \, x^{3} + 28 \, x^{2} + 12 \, x + 8\right )} e^{4} + 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{4 \, x^{3}} + \frac {{\left ({\left (13 \, x^{3} + 28 \, x^{2} + 12 \, x + 8\right )} e^{4} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{4 \, x^{3}} + 4\right )} \] Input:
integrate(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*( (12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+ 4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x, algorithm="fricas")
Output:
e^(1/4*(4*x^3*e^(-1/4*((13*x^3 + 28*x^2 + 12*x + 8)*e^4 - 4*(3*x^2 + x + 1 )*e^(x + 4))*e^(-4)/x^3 + 4) - (13*x^3 + 28*x^2 + 12*x + 8)*e^4 + 4*(3*x^2 + x + 1)*e^(x + 4))*e^(-4)/x^3 + 1/4*((13*x^3 + 28*x^2 + 12*x + 8)*e^4 - 4*(3*x^2 + x + 1)*e^(x + 4))*e^(-4)/x^3 + 4)
Time = 1.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx=e^{4} e^{e^{\frac {- \frac {13 x^{3}}{4} - 7 x^{2} - 3 x + \frac {\left (12 x^{2} + 4 x + 4\right ) e^{x}}{4} - 2}{x^{3}}}} \] Input:
integrate(((3*x**3-2*x**2-x-3)*exp(4)*exp(x)+(7*x**2+6*x+6)*exp(4))*exp(1/ 4*((12*x**2+4*x+4)*exp(x)-13*x**3-28*x**2-12*x-8)/x**3)*exp(exp(1/4*((12*x **2+4*x+4)*exp(x)-13*x**3-28*x**2-12*x-8)/x**3))/x**4,x)
Output:
exp(4)*exp(exp((-13*x**3/4 - 7*x**2 - 3*x + (12*x**2 + 4*x + 4)*exp(x)/4 - 2)/x**3))
Time = 0.74 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18 \[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx=e^{\left (e^{\left (\frac {3 \, e^{x}}{x} - \frac {7}{x} + \frac {e^{x}}{x^{2}} - \frac {3}{x^{2}} + \frac {e^{x}}{x^{3}} - \frac {2}{x^{3}} - \frac {13}{4}\right )} + 4\right )} \] Input:
integrate(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*( (12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+ 4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x, algorithm="maxima")
Output:
e^(e^(3*e^x/x - 7/x + e^x/x^2 - 3/x^2 + e^x/x^3 - 2/x^3 - 13/4) + 4)
\[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx=\int { \frac {{\left ({\left (7 \, x^{2} + 6 \, x + 6\right )} e^{4} + {\left (3 \, x^{3} - 2 \, x^{2} - x - 3\right )} e^{\left (x + 4\right )}\right )} e^{\left (-\frac {13 \, x^{3} + 28 \, x^{2} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{x} + 12 \, x + 8}{4 \, x^{3}} + e^{\left (-\frac {13 \, x^{3} + 28 \, x^{2} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{x} + 12 \, x + 8}{4 \, x^{3}}\right )}\right )}}{x^{4}} \,d x } \] Input:
integrate(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*( (12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+ 4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x, algorithm="giac")
Output:
integrate(((7*x^2 + 6*x + 6)*e^4 + (3*x^3 - 2*x^2 - x - 3)*e^(x + 4))*e^(- 1/4*(13*x^3 + 28*x^2 - 4*(3*x^2 + x + 1)*e^x + 12*x + 8)/x^3 + e^(-1/4*(13 *x^3 + 28*x^2 - 4*(3*x^2 + x + 1)*e^x + 12*x + 8)/x^3))/x^4, x)
Time = 3.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx={\mathrm {e}}^4\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {13}{4}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x^2}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x^3}}\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{-\frac {3}{x^2}}\,{\mathrm {e}}^{-\frac {2}{x^3}}\,{\mathrm {e}}^{-\frac {7}{x}}} \] Input:
int((exp(exp(-(3*x - (exp(x)*(4*x + 12*x^2 + 4))/4 + 7*x^2 + (13*x^3)/4 + 2)/x^3))*exp(-(3*x - (exp(x)*(4*x + 12*x^2 + 4))/4 + 7*x^2 + (13*x^3)/4 + 2)/x^3)*(exp(4)*(6*x + 7*x^2 + 6) - exp(4)*exp(x)*(x + 2*x^2 - 3*x^3 + 3)) )/x^4,x)
Output:
exp(4)*exp(exp(-13/4)*exp(exp(x)/x^2)*exp(exp(x)/x^3)*exp((3*exp(x))/x)*ex p(-3/x^2)*exp(-2/x^3)*exp(-7/x))
\[ \int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}} \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx=\int \frac {\left (\left (3 x^{3}-2 x^{2}-x -3\right ) {\mathrm e}^{4} {\mathrm e}^{x}+\left (7 x^{2}+6 x +6\right ) {\mathrm e}^{4}\right ) {\mathrm e}^{\frac {\left (12 x^{2}+4 x +4\right ) {\mathrm e}^{x}-13 x^{3}-28 x^{2}-12 x -8}{4 x^{3}}} {\mathrm e}^{{\mathrm e}^{\frac {\left (12 x^{2}+4 x +4\right ) {\mathrm e}^{x}-13 x^{3}-28 x^{2}-12 x -8}{4 x^{3}}}}}{x^{4}}d x \] Input:
int(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*((12*x^ 2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+4)*exp (x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x)
Output:
int(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*((12*x^ 2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+4)*exp (x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x)