Integrand size = 219, antiderivative size = 35 \[ \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx=e^{-\frac {x}{-e^2+\frac {e^{2 e^x}}{x}+2 x}} x-3 x^3 \]
Time = 0.46 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx=e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} x-3 x^3 \]
Integrate[(E^(4*E^x) + E^4*x^2 - 3*E^2*x^3 + 4*x^4 + E^(2*E^x)*(-2*E^2*x + 2*x^2 + 2*E^x*x^3) + E^(x^2/(E^(2*E^x) - E^2*x + 2*x^2))*(-9*E^(4*E^x)*x^ 2 - 9*E^4*x^4 + 36*E^2*x^5 - 36*x^6 + E^(2*E^x)*(18*E^2*x^3 - 36*x^4)))/(E ^(x^2/(E^(2*E^x) - E^2*x + 2*x^2))*(E^(4*E^x) + E^4*x^2 - 4*E^2*x^3 + 4*x^ 4 + E^(2*E^x)*(-2*E^2*x + 4*x^2))),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} \left (4 x^4-3 e^2 x^3+e^4 x^2+e^{2 e^x} \left (2 e^x x^3+2 x^2-2 e^2 x\right )+e^{\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} \left (-36 x^6+36 e^2 x^5-9 e^4 x^4-9 e^{4 e^x} x^2+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )+e^{4 e^x}\right )}{4 x^4-4 e^2 x^3+e^4 x^2+e^{2 e^x} \left (4 x^2-2 e^2 x\right )+e^{4 e^x}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} \left (4 x^4-3 e^2 x^3+e^4 x^2+e^{2 e^x} \left (2 e^x x^3+2 x^2-2 e^2 x\right )+e^{\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} \left (-36 x^6+36 e^2 x^5-9 e^4 x^4-9 e^{4 e^x} x^2+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )+e^{4 e^x}\right )}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{4-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} x^2}{\left (-2 x^2+e^2 x-e^{2 e^x}\right )^2}-9 x^2+\frac {2 e^{2 e^x-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} \left (e^x x^2+x-e^2\right ) x}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}+\frac {e^{4 e^x-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}}}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}+\frac {4 e^{-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} x^4}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}-\frac {3 e^{2-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} x^3}{\left (-2 x^2+e^2 x-e^{2 e^x}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {e^{-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}+2 e^x+2} x}{\left (-2 x^2+e^2 x-e^{2 e^x}\right )^2}dx+\int \frac {e^{4-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} x^2}{\left (-2 x^2+e^2 x-e^{2 e^x}\right )^2}dx+\int \frac {e^{4 e^x-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}}}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}dx+2 \int \frac {e^{2 e^x-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} x^2}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}dx+4 \int \frac {e^{-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} x^4}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}dx-3 \int \frac {e^{2-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}} x^3}{\left (-2 x^2+e^2 x-e^{2 e^x}\right )^2}dx+2 \int \frac {e^{-\frac {x^2}{2 x^2-e^2 x+e^{2 e^x}}+x+2 e^x} x^3}{\left (2 x^2-e^2 x+e^{2 e^x}\right )^2}dx-3 x^3\) |
Int[(E^(4*E^x) + E^4*x^2 - 3*E^2*x^3 + 4*x^4 + E^(2*E^x)*(-2*E^2*x + 2*x^2 + 2*E^x*x^3) + E^(x^2/(E^(2*E^x) - E^2*x + 2*x^2))*(-9*E^(4*E^x)*x^2 - 9* E^4*x^4 + 36*E^2*x^5 - 36*x^6 + E^(2*E^x)*(18*E^2*x^3 - 36*x^4)))/(E^(x^2/ (E^(2*E^x) - E^2*x + 2*x^2))*(E^(4*E^x) + E^4*x^2 - 4*E^2*x^3 + 4*x^4 + E^ (2*E^x)*(-2*E^2*x + 4*x^2))),x]
3.7.34.3.1 Defintions of rubi rules used
Time = 89.72 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-3 x^{3}+x \,{\mathrm e}^{\frac {x^{2}}{-{\mathrm e}^{2 \,{\mathrm e}^{x}}+{\mathrm e}^{2} x -2 x^{2}}}\) | \(33\) |
| parallelrisch | \(\frac {\left (-3 \,{\mathrm e}^{2} x^{4} {\mathrm e}^{-\frac {x^{2}}{-{\mathrm e}^{2 \,{\mathrm e}^{x}}+{\mathrm e}^{2} x -2 x^{2}}}+6 x^{5} {\mathrm e}^{-\frac {x^{2}}{-{\mathrm e}^{2 \,{\mathrm e}^{x}}+{\mathrm e}^{2} x -2 x^{2}}}+3 \,{\mathrm e}^{-\frac {x^{2}}{-{\mathrm e}^{2 \,{\mathrm e}^{x}}+{\mathrm e}^{2} x -2 x^{2}}} {\mathrm e}^{2 \,{\mathrm e}^{x}} x^{3}+x^{2} {\mathrm e}^{2}-2 x^{3}-x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}\right ) {\mathrm e}^{\frac {x^{2}}{-{\mathrm e}^{2 \,{\mathrm e}^{x}}+{\mathrm e}^{2} x -2 x^{2}}}}{-{\mathrm e}^{2 \,{\mathrm e}^{x}}+{\mathrm e}^{2} x -2 x^{2}}\) | \(165\) |
int(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^4*exp( 2)^2+36*exp(2)*x^5-36*x^6)*exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2))+exp(exp (x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-3*x^3*ex p(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp(2)^2- 4*x^3*exp(2)+4*x^4)/exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2)),x,method=_RETU RNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx=-{\left (3 \, x^{3} e^{\left (\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} - x\right )} e^{\left (-\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} \]
integrate(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^ 4*exp(2)^2+36*exp(2)*x^5-36*x^6)*exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2))+e xp(exp(x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-3* x^3*exp(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp (2)^2-4*x^3*exp(2)+4*x^4)/exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2)),x, algor ithm=\
Time = 1.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx=- 3 x^{3} + x e^{- \frac {x^{2}}{2 x^{2} - x e^{2} + e^{2 e^{x}}}} \]
integrate(((-9*x**2*exp(exp(x))**4+(18*x**3*exp(2)-36*x**4)*exp(exp(x))**2 -9*x**4*exp(2)**2+36*exp(2)*x**5-36*x**6)*exp(x**2/(exp(exp(x))**2-exp(2)* x+2*x**2))+exp(exp(x))**4+(2*exp(x)*x**3-2*exp(2)*x+2*x**2)*exp(exp(x))**2 +x**2*exp(2)**2-3*x**3*exp(2)+4*x**4)/(exp(exp(x))**4+(-2*exp(2)*x+4*x**2) *exp(exp(x))**2+x**2*exp(2)**2-4*x**3*exp(2)+4*x**4)/exp(x**2/(exp(exp(x)) **2-exp(2)*x+2*x**2)),x)
\[ \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx=\int { \frac {{\left (4 \, x^{4} - 3 \, x^{3} e^{2} + x^{2} e^{4} - 9 \, {\left (4 \, x^{6} - 4 \, x^{5} e^{2} + x^{4} e^{4} + x^{2} e^{\left (4 \, e^{x}\right )} + 2 \, {\left (2 \, x^{4} - x^{3} e^{2}\right )} e^{\left (2 \, e^{x}\right )}\right )} e^{\left (\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} + 2 \, {\left (x^{3} e^{x} + x^{2} - x e^{2}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (4 \, e^{x}\right )}\right )} e^{\left (-\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )}}{4 \, x^{4} - 4 \, x^{3} e^{2} + x^{2} e^{4} + 2 \, {\left (2 \, x^{2} - x e^{2}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (4 \, e^{x}\right )}} \,d x } \]
integrate(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^ 4*exp(2)^2+36*exp(2)*x^5-36*x^6)*exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2))+e xp(exp(x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-3* x^3*exp(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp (2)^2-4*x^3*exp(2)+4*x^4)/exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2)),x, algor ithm=\
-3*x^3 + integrate((4*x^4 - 3*x^3*e^2 + x^2*e^4 + 2*(x^3*e^x + x^2 - x*e^2 )*e^(2*e^x) + e^(4*e^x))*e^(-1/2*x*e^2/(2*x^2 - x*e^2 + e^(2*e^x)) + 1/2*e ^(2*e^x)/(2*x^2 - x*e^2 + e^(2*e^x)))/(4*x^4*e^(1/2) - 4*x^3*e^(5/2) + x^2 *e^(9/2) + 2*(2*x^2*e^(1/2) - x*e^(5/2))*e^(2*e^x) + e^(4*e^x + 1/2)), x)
Time = 0.61 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx=-{\left (3 \, x^{3} e^{\left (\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} - x\right )} e^{\left (-\frac {x^{2}}{2 \, x^{2} - x e^{2} + e^{\left (2 \, e^{x}\right )}}\right )} \]
integrate(((-9*x^2*exp(exp(x))^4+(18*x^3*exp(2)-36*x^4)*exp(exp(x))^2-9*x^ 4*exp(2)^2+36*exp(2)*x^5-36*x^6)*exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2))+e xp(exp(x))^4+(2*exp(x)*x^3-2*exp(2)*x+2*x^2)*exp(exp(x))^2+x^2*exp(2)^2-3* x^3*exp(2)+4*x^4)/(exp(exp(x))^4+(-2*exp(2)*x+4*x^2)*exp(exp(x))^2+x^2*exp (2)^2-4*x^3*exp(2)+4*x^4)/exp(x^2/(exp(exp(x))^2-exp(2)*x+2*x^2)),x, algor ithm=\
Time = 9.80 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (e^{4 e^x}+e^4 x^2-3 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+2 x^2+2 e^x x^3\right )+e^{\frac {x^2}{e^{2 e^x}-e^2 x+2 x^2}} \left (-9 e^{4 e^x} x^2-9 e^4 x^4+36 e^2 x^5-36 x^6+e^{2 e^x} \left (18 e^2 x^3-36 x^4\right )\right )\right )}{e^{4 e^x}+e^4 x^2-4 e^2 x^3+4 x^4+e^{2 e^x} \left (-2 e^2 x+4 x^2\right )} \, dx=x\,{\mathrm {e}}^{-\frac {x^2}{{\mathrm {e}}^{2\,{\mathrm {e}}^x}-x\,{\mathrm {e}}^2+2\,x^2}}-3\,x^3 \]
int((exp(-x^2/(exp(2*exp(x)) - x*exp(2) + 2*x^2))*(exp(4*exp(x)) - 3*x^3*e xp(2) + x^2*exp(4) - exp(x^2/(exp(2*exp(x)) - x*exp(2) + 2*x^2))*(9*x^4*ex p(4) - 36*x^5*exp(2) - exp(2*exp(x))*(18*x^3*exp(2) - 36*x^4) + 36*x^6 + 9 *x^2*exp(4*exp(x))) + 4*x^4 + exp(2*exp(x))*(2*x^3*exp(x) - 2*x*exp(2) + 2 *x^2)))/(exp(4*exp(x)) - 4*x^3*exp(2) + x^2*exp(4) - exp(2*exp(x))*(2*x*ex p(2) - 4*x^2) + 4*x^4),x)