Integrand size = 220, antiderivative size = 38 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=\frac {1}{4} x \left (x+\frac {x^2}{\left (-e^{-e^x+e^{(2-x) x}}+\frac {x}{e^2}\right )^2}\right ) \] Output:
1/4*(x^2/(x/exp(2)-exp(exp((2-x)*x)-exp(x)))^2+x)*x
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=\frac {1}{4} \left (x^2+\frac {e^{4+2 e^x} x^3}{\left (e^{2+e^{-((-2+x) x)}}-e^{e^x} x\right )^2}\right ) \] Input:
Integrate[(2*E^(6 - 3*E^x + 3*E^(2*x - x^2))*x - 6*E^(4 - 2*E^x + 2*E^(2*x - x^2))*x^2 - E^4*x^3 - 2*x^4 + E^(-E^x + E^(2*x - x^2))*(3*E^6*x^2 + 6*E ^2*x^3 + 2*E^(6 + x)*x^3 + E^(6 + 2*x - x^2)*(-4*x^3 + 4*x^4)))/(4*E^(6 - 3*E^x + 3*E^(2*x - x^2)) - 12*E^(4 - 2*E^x + 2*E^(2*x - x^2))*x + 12*E^(2 - E^x + E^(2*x - x^2))*x^2 - 4*x^3),x]
Output:
(x^2 + (E^(4 + 2*E^x)*x^3)/(E^(2 + E^(-((-2 + x)*x))) - E^E^x*x)^2)/4
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.
| \(\displaystyle \int \frac {-2 x^4-e^4 x^3-6 e^{2 e^{2 x-x^2}-2 e^x+4} x^2+2 e^{3 e^{2 x-x^2}-3 e^x+6} x+e^{e^{2 x-x^2}-e^x} \left (2 e^{x+6} x^3+6 e^2 x^3+3 e^6 x^2+e^{-x^2+2 x+6} \left (4 x^4-4 x^3\right )\right )}{-4 x^3+12 e^{e^{2 x-x^2}-e^x+2} x^2-12 e^{2 e^{2 x-x^2}-2 e^x+4} x+4 e^{3 e^{2 x-x^2}-3 e^x+6}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{3 e^x} \left (-2 x^4-e^4 x^3-6 e^{2 e^{2 x-x^2}-2 e^x+4} x^2+2 e^{3 e^{2 x-x^2}-3 e^x+6} x+e^{e^{2 x-x^2}-e^x} \left (2 e^{x+6} x^3+6 e^2 x^3+3 e^6 x^2+e^{-x^2+2 x+6} \left (4 x^4-4 x^3\right )\right )\right )}{4 \left (e^{e^{-((x-2) x)}+2}-e^{e^x} x\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {e^{3 e^x} \left (-2 x^4-e^4 x^3-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2+2 e^{6-3 e^x+3 e^{2 x-x^2}} x+e^{-e^x+e^{2 x-x^2}} \left (2 e^{x+6} x^3+6 e^2 x^3+3 e^6 x^2-4 e^{-x^2+2 x+6} \left (x^3-x^4\right )\right )\right )}{\left (e^{2+e^{(2-x) x}}-e^{e^x} x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {2 e^{3 e^x} x^4}{\left (e^{e^x} x-e^{2+e^{-((x-2) x)}}\right )^3}-\frac {e^{4+3 e^x} x^3}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}+\frac {6 e^{2+2 e^x+e^{-((x-2) x)}} x^3}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}+\frac {2 e^{x+2 e^x+e^{-((x-2) x)}+6} x^3}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}+\frac {4 e^{-x^2+2 x+2 e^x+e^{-((x-2) x)}+6} (x-1) x^3}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}+\frac {3 e^{6+2 e^x+e^{-((x-2) x)}} x^2}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}-\frac {6 e^{4+e^x+2 e^{-((x-2) x)}} x^2}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}+\frac {2 e^{6+3 e^{-((x-2) x)}} x}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{4} \int \frac {e^{-x^2} x \left (-2 e^{x^2+3 e^x} x^3-e^{x^2+3 e^x+4} x^2+6 e^{x^2+2 e^x+e^{-((x-2) x)}+2} x^2+2 e^{x^2+x+2 e^x+e^{-((x-2) x)}+6} x^2+4 e^{2 x+2 e^x+e^{-((x-2) x)}+6} (x-1) x^2+3 e^{x^2+2 e^x+e^{-((x-2) x)}+6} x-6 e^{x^2+e^x+2 e^{-((x-2) x)}+4} x+2 e^{x^2+3 e^{-((x-2) x)}+6}\right )}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {4 e^{-x^2+2 x+2 e^x+e^{-((x-2) x)}+6} (x-1) x^3}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}+\frac {\left (2 e^{3 e^x} x^3+e^{4+3 e^x} x^2-6 e^{2+2 e^x+e^{-((x-2) x)}} x^2-2 e^{x+2 e^x+e^{-((x-2) x)}+6} x^2-3 e^{6+2 e^x+e^{-((x-2) x)}} x+6 e^{4+e^x+2 e^{-((x-2) x)}} x-2 e^{6+3 e^{-((x-2) x)}}\right ) x}{\left (e^{e^x} x-e^{2+e^{-((x-2) x)}}\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {4 e^{-x^2+2 x+2 e^x+e^{-((x-2) x)}+6} (x-1) x^3}{\left (e^{2+e^{-((x-2) x)}}-e^{e^x} x\right )^3}+\frac {\left (2 e^{3 e^x} x^3+e^{4+3 e^x} x^2-6 e^{2+2 e^x+e^{-((x-2) x)}} x^2-2 e^{x+2 e^x+e^{-((x-2) x)}+6} x^2-3 e^{6+2 e^x+e^{-((x-2) x)}} x+6 e^{4+e^x+2 e^{-((x-2) x)}} x-2 e^{6+3 e^{-((x-2) x)}}\right ) x}{\left (e^{e^x} x-e^{2+e^{-((x-2) x)}}\right )^3}\right )dx\) |
Input:
Int[(2*E^(6 - 3*E^x + 3*E^(2*x - x^2))*x - 6*E^(4 - 2*E^x + 2*E^(2*x - x^2 ))*x^2 - E^4*x^3 - 2*x^4 + E^(-E^x + E^(2*x - x^2))*(3*E^6*x^2 + 6*E^2*x^3 + 2*E^(6 + x)*x^3 + E^(6 + 2*x - x^2)*(-4*x^3 + 4*x^4)))/(4*E^(6 - 3*E^x + 3*E^(2*x - x^2)) - 12*E^(4 - 2*E^x + 2*E^(2*x - x^2))*x + 12*E^(2 - E^x + E^(2*x - x^2))*x^2 - 4*x^3),x]
Output:
$Aborted
Time = 3.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(\frac {x^{2}}{4}+\frac {x^{3} {\mathrm e}^{4}}{4 \left ({\mathrm e}^{-{\mathrm e}^{x}+{\mathrm e}^{-\left (-2+x \right ) x}+2}-x \right )^{2}}\) | \(34\) |
| parallelrisch | \(\frac {x^{4}+x^{3} {\mathrm e}^{4}-2 \,{\mathrm e}^{2} {\mathrm e}^{-{\mathrm e}^{x}+{\mathrm e}^{-x^{2}+2 x}} x^{3}+x^{2} {\mathrm e}^{4} {\mathrm e}^{-2 \,{\mathrm e}^{x}+2 \,{\mathrm e}^{-x^{2}+2 x}}}{4 \,{\mathrm e}^{4} {\mathrm e}^{-2 \,{\mathrm e}^{x}+2 \,{\mathrm e}^{-x^{2}+2 x}}-8 x \,{\mathrm e}^{2} {\mathrm e}^{-{\mathrm e}^{x}+{\mathrm e}^{-x^{2}+2 x}}+4 x^{2}}\) | \(114\) |
Input:
int((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-exp(x)+ exp(-x^2+2*x))^2+(2*x^3*exp(2)^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(-x^2+2* x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp(2)^2-2* x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp(x)+ex p(-x^2+2*x))^2+12*x^2*exp(2)*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x,method=_R ETURNVERBOSE)
Output:
1/4*x^2+1/4*x^3*exp(4)/(exp(-exp(x)+exp(-(-2+x)*x)+2)-x)^2
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (31) = 62\).
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.58 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=\frac {x^{4} + x^{3} e^{4} - 2 \, x^{3} e^{\left ({\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )} + x^{2} e^{\left (2 \, {\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )}}{4 \, {\left (x^{2} - 2 \, x e^{\left ({\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )} + e^{\left (2 \, {\left (2 \, e^{6} + e^{\left (-x^{2} + 2 \, x + 6\right )} - e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}\right )}\right )}} \] Input:
integrate((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-e xp(x)+exp(-x^2+2*x))^2+(2*x^3*exp(2)^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(- x^2+2*x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp(2 )^2-2*x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp (x)+exp(-x^2+2*x))^2+12*x^2*exp(2)*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x, al gorithm="fricas")
Output:
1/4*(x^4 + x^3*e^4 - 2*x^3*e^((2*e^6 + e^(-x^2 + 2*x + 6) - e^(x + 6))*e^( -6)) + x^2*e^(2*(2*e^6 + e^(-x^2 + 2*x + 6) - e^(x + 6))*e^(-6)))/(x^2 - 2 *x*e^((2*e^6 + e^(-x^2 + 2*x + 6) - e^(x + 6))*e^(-6)) + e^(2*(2*e^6 + e^( -x^2 + 2*x + 6) - e^(x + 6))*e^(-6)))
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.53 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=\frac {x^{3} e^{4}}{4 x^{2} - 8 x e^{2} e^{- e^{x} + e^{- x^{2} + 2 x}} + 4 e^{4} e^{- 2 e^{x} + 2 e^{- x^{2} + 2 x}}} + \frac {x^{2}}{4} \] Input:
integrate((2*x*exp(2)**3*exp(-exp(x)+exp(-x**2+2*x))**3-6*x**2*exp(2)**2*e xp(-exp(x)+exp(-x**2+2*x))**2+(2*x**3*exp(2)**3*exp(x)+(4*x**4-4*x**3)*exp (2)**3*exp(-x**2+2*x)+3*x**2*exp(2)**3+6*x**3*exp(2))*exp(-exp(x)+exp(-x** 2+2*x))-x**3*exp(2)**2-2*x**4)/(4*exp(2)**3*exp(-exp(x)+exp(-x**2+2*x))**3 -12*x*exp(2)**2*exp(-exp(x)+exp(-x**2+2*x))**2+12*x**2*exp(2)*exp(-exp(x)+ exp(-x**2+2*x))-4*x**3),x)
Output:
x**3*exp(4)/(4*x**2 - 8*x*exp(2)*exp(-exp(x) + exp(-x**2 + 2*x)) + 4*exp(4 )*exp(-2*exp(x) + 2*exp(-x**2 + 2*x))) + x**2/4
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (31) = 62\).
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.76 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=-\frac {2 \, x^{3} e^{\left (e^{\left (-x^{2} + 2 \, x\right )} + e^{x} + 2\right )} - x^{2} e^{\left (2 \, e^{\left (-x^{2} + 2 \, x\right )} + 4\right )} - {\left (x^{4} + x^{3} e^{4}\right )} e^{\left (2 \, e^{x}\right )}}{4 \, {\left (x^{2} e^{\left (2 \, e^{x}\right )} - 2 \, x e^{\left (e^{\left (-x^{2} + 2 \, x\right )} + e^{x} + 2\right )} + e^{\left (2 \, e^{\left (-x^{2} + 2 \, x\right )} + 4\right )}\right )}} \] Input:
integrate((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-e xp(x)+exp(-x^2+2*x))^2+(2*x^3*exp(2)^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(- x^2+2*x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp(2 )^2-2*x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp (x)+exp(-x^2+2*x))^2+12*x^2*exp(2)*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x, al gorithm="maxima")
Output:
-1/4*(2*x^3*e^(e^(-x^2 + 2*x) + e^x + 2) - x^2*e^(2*e^(-x^2 + 2*x) + 4) - (x^4 + x^3*e^4)*e^(2*e^x))/(x^2*e^(2*e^x) - 2*x*e^(e^(-x^2 + 2*x) + e^x + 2) + e^(2*e^(-x^2 + 2*x) + 4))
Leaf count of result is larger than twice the leaf count of optimal. 14409 vs. \(2 (31) = 62\).
Time = 1.67 (sec) , antiderivative size = 14409, normalized size of antiderivative = 379.18 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=\text {Too large to display} \] Input:
integrate((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-e xp(x)+exp(-x^2+2*x))^2+(2*x^3*exp(2)^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(- x^2+2*x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp(2 )^2-2*x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp (x)+exp(-x^2+2*x))^2+12*x^2*exp(2)*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x, al gorithm="giac")
Output:
1/4*(8*x^16*e^(-x^2 + 6*x + 4*e^(-x^2 + 2*x) + 2*e^x) - 64*x^15*e^(-x^2 + 6*x + 5*e^(-x^2 + 2*x) + e^x + 2) + 8*x^15*e^(-x^2 + 6*x + 4*e^(-x^2 + 2*x ) + 2*e^x + 4) - 24*x^15*e^(-x^2 + 6*x + 4*e^(-x^2 + 2*x) + 2*e^x) + 12*x^ 15*e^(5*x + 4*e^(-x^2 + 2*x) + 2*e^x) + 6*x^14*e^(x^2 + 4*x + 4*e^(-x^2 + 2*x) + 2*e^x) + 224*x^14*e^(-x^2 + 6*x + 6*e^(-x^2 + 2*x) + 4) - 48*x^14*e ^(-x^2 + 6*x + 5*e^(-x^2 + 2*x) + e^x + 6) + 192*x^14*e^(-x^2 + 6*x + 5*e^ (-x^2 + 2*x) + e^x + 2) - 24*x^14*e^(-x^2 + 6*x + 4*e^(-x^2 + 2*x) + 2*e^x + 4) + 24*x^14*e^(-x^2 + 6*x + 4*e^(-x^2 + 2*x) + 2*e^x) - 96*x^14*e^(5*x + 5*e^(-x^2 + 2*x) + e^x + 2) + 12*x^14*e^(5*x + 4*e^(-x^2 + 2*x) + 2*e^x + 4) - 24*x^14*e^(5*x + 4*e^(-x^2 + 2*x) + 2*e^x) + 12*x^14*e^(4*x + 4*e^ (-x^2 + 2*x) + 2*e^x) + x^13*e^(2*x^2 + 3*x + 4*e^(-x^2 + 2*x) + 2*e^x) - 48*x^13*e^(x^2 + 4*x + 5*e^(-x^2 + 2*x) + e^x + 2) + 6*x^13*e^(x^2 + 4*x + 4*e^(-x^2 + 2*x) + 2*e^x + 4) - 6*x^13*e^(x^2 + 4*x + 4*e^(-x^2 + 2*x) + 2*e^x) + 12*x^13*e^(x^2 + 3*x + 4*e^(-x^2 + 2*x) + 2*e^x) - 448*x^13*e^(-x ^2 + 6*x + 7*e^(-x^2 + 2*x) - e^x + 6) + 120*x^13*e^(-x^2 + 6*x + 6*e^(-x^ 2 + 2*x) + 8) - 672*x^13*e^(-x^2 + 6*x + 6*e^(-x^2 + 2*x) + 4) + 144*x^13* e^(-x^2 + 6*x + 5*e^(-x^2 + 2*x) + e^x + 6) - 192*x^13*e^(-x^2 + 6*x + 5*e ^(-x^2 + 2*x) + e^x + 2) + 24*x^13*e^(-x^2 + 6*x + 4*e^(-x^2 + 2*x) + 2*e^ x + 4) - 8*x^13*e^(-x^2 + 6*x + 4*e^(-x^2 + 2*x) + 2*e^x) + 336*x^13*e^(5* x + 6*e^(-x^2 + 2*x) + 4) - 72*x^13*e^(5*x + 5*e^(-x^2 + 2*x) + e^x + 6...
Time = 3.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.68 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=\frac {x^2\,\left (x+{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-2\,{\mathrm {e}}^x}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-{\mathrm {e}}^x-2}+x^2\,{\mathrm {e}}^{-4}\right )}{4\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-2\,{\mathrm {e}}^x}-2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-x^2}-{\mathrm {e}}^x-2}+x^2\,{\mathrm {e}}^{-4}\right )} \] Input:
int((x^3*exp(4) - exp(exp(2*x - x^2) - exp(x))*(6*x^3*exp(2) + 3*x^2*exp(6 ) - exp(6)*exp(2*x - x^2)*(4*x^3 - 4*x^4) + 2*x^3*exp(6)*exp(x)) + 2*x^4 - 2*x*exp(3*exp(2*x - x^2) - 3*exp(x))*exp(6) + 6*x^2*exp(2*exp(2*x - x^2) - 2*exp(x))*exp(4))/(4*x^3 - 4*exp(3*exp(2*x - x^2) - 3*exp(x))*exp(6) + 1 2*x*exp(2*exp(2*x - x^2) - 2*exp(x))*exp(4) - 12*x^2*exp(2)*exp(exp(2*x - x^2) - exp(x))),x)
Output:
(x^2*(x + exp(2*exp(2*x)*exp(-x^2) - 2*exp(x)) - 2*x*exp(exp(2*x)*exp(-x^2 ) - exp(x) - 2) + x^2*exp(-4)))/(4*(exp(2*exp(2*x)*exp(-x^2) - 2*exp(x)) - 2*x*exp(exp(2*x)*exp(-x^2) - exp(x) - 2) + x^2*exp(-4)))
Time = 0.45 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.74 \[ \int \frac {2 e^{6-3 e^x+3 e^{2 x-x^2}} x-6 e^{4-2 e^x+2 e^{2 x-x^2}} x^2-e^4 x^3-2 x^4+e^{-e^x+e^{2 x-x^2}} \left (3 e^6 x^2+6 e^2 x^3+2 e^{6+x} x^3+e^{6+2 x-x^2} \left (-4 x^3+4 x^4\right )\right )}{4 e^{6-3 e^x+3 e^{2 x-x^2}}-12 e^{4-2 e^x+2 e^{2 x-x^2}} x+12 e^{2-e^x+e^{2 x-x^2}} x^2-4 x^3} \, dx=\frac {x^{2} \left (e^{2 e^{x}} e^{4} x +e^{2 e^{x}} x^{2}-2 e^{\frac {e^{x^{2}+x}+e^{2 x}}{e^{x^{2}}}} e^{2} x +e^{\frac {2 e^{2 x}}{e^{x^{2}}}} e^{4}\right )}{4 e^{2 e^{x}} x^{2}-8 e^{\frac {e^{x^{2}+x}+e^{2 x}}{e^{x^{2}}}} e^{2} x +4 e^{\frac {2 e^{2 x}}{e^{x^{2}}}} e^{4}} \] Input:
int((2*x*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-6*x^2*exp(2)^2*exp(-exp(x)+ exp(-x^2+2*x))^2+(2*x^3*exp(2)^3*exp(x)+(4*x^4-4*x^3)*exp(2)^3*exp(-x^2+2* x)+3*x^2*exp(2)^3+6*x^3*exp(2))*exp(-exp(x)+exp(-x^2+2*x))-x^3*exp(2)^2-2* x^4)/(4*exp(2)^3*exp(-exp(x)+exp(-x^2+2*x))^3-12*x*exp(2)^2*exp(-exp(x)+ex p(-x^2+2*x))^2+12*x^2*exp(2)*exp(-exp(x)+exp(-x^2+2*x))-4*x^3),x)
Output:
(x**2*(e**(2*e**x)*e**4*x + e**(2*e**x)*x**2 - 2*e**((e**(x**2 + x) + e**( 2*x))/e**(x**2))*e**2*x + e**((2*e**(2*x))/e**(x**2))*e**4))/(4*(e**(2*e** x)*x**2 - 2*e**((e**(x**2 + x) + e**(2*x))/e**(x**2))*e**2*x + e**((2*e**( 2*x))/e**(x**2))*e**4))