Integrand size = 122, antiderivative size = 33 \[ \int \frac {e^{e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2} \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {e^{-5+x-3 e^{-x^2} x} \left (4 e^{-x}+x\right )}{(-1+x)^2 x} \]
Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {e^{e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2} \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {e^{-5-3 e^{-x^2} x} \left (4+e^x x\right )}{(-1+x)^2 x} \]
Integrate[(E^((E^x^2*(-5 + x) - 3*x)/E^x^2 - x - x^2)*(12*x - 12*x^2 - 24* x^3 + 24*x^4 + E^x*(3*x^2 - 3*x^3 - 6*x^4 + 6*x^5) + E^x^2*(4 - 12*x + E^x *(-3*x^2 + x^3))))/(-x^2 + 3*x^3 - 3*x^4 + x^5),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.
| \(\displaystyle \int \frac {\left (24 x^4-24 x^3-12 x^2+e^{x^2} \left (e^x \left (x^3-3 x^2\right )-12 x+4\right )+e^x \left (6 x^5-6 x^4-3 x^3+3 x^2\right )+12 x\right ) \exp \left (-x^2+e^{-x^2} \left (e^{x^2} (x-5)-3 x\right )-x\right )}{x^5-3 x^4+3 x^3-x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (24 x^4-24 x^3-12 x^2+e^{x^2} \left (e^x \left (x^3-3 x^2\right )-12 x+4\right )+e^x \left (6 x^5-6 x^4-3 x^3+3 x^2\right )+12 x\right ) \exp \left (-x^2+e^{-x^2} \left (e^{x^2} (x-5)-3 x\right )-x\right )}{x^2 \left (x^3-3 x^2+3 x-1\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (24 x^4-24 x^3-12 x^2+e^{x^2} \left (e^x \left (x^3-3 x^2\right )-12 x+4\right )+e^x \left (6 x^5-6 x^4-3 x^3+3 x^2\right )+12 x\right ) \exp \left (-x^2+e^{-x^2} \left (e^{x^2} (x-5)-3 x\right )-x\right )}{(x-1)^3 x^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{-e^{-x^2} \left (e^{x^2} x^2+5 e^{x^2}+3 x\right )} \left (-24 x^4+24 x^3+12 x^2-e^{x^2} \left (e^x \left (x^3-3 x^2\right )-12 x+4\right )-e^x \left (6 x^5-6 x^4-3 x^3+3 x^2\right )-12 x\right )}{(1-x)^3 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {24 e^{-e^{-x^2} \left (e^{x^2} x^2+5 e^{x^2}+3 x\right )} x^2}{(x-1)^3}-\frac {24 e^{-e^{-x^2} \left (e^{x^2} x^2+5 e^{x^2}+3 x\right )} x}{(x-1)^3}+\frac {3 e^{x-e^{-x^2} \left (e^{x^2} x^2+5 e^{x^2}+3 x\right )} \left (2 x^2-1\right )}{(x-1)^2}-\frac {12 e^{-e^{-x^2} \left (e^{x^2} x^2+5 e^{x^2}+3 x\right )}}{(x-1)^3}+\frac {12 e^{-e^{-x^2} \left (e^{x^2} x^2+5 e^{x^2}+3 x\right )}}{(x-1)^3 x}+\frac {e^{x^2-e^{-x^2} \left (e^{x^2} x^2+5 e^{x^2}+3 x\right )} \left (e^x x^3-3 e^x x^2-12 x+4\right )}{(x-1)^3 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \int e^{x-e^{-x^2} \left (e^{x^2} x^2+3 x+5 e^{x^2}\right )}dx-8 \int \frac {e^{-e^{-x^2} \left (3 x+5 e^{x^2}\right )}}{(x-1)^3}dx-2 \int \frac {e^{x-e^{-x^2} \left (3 x+5 e^{x^2}\right )}}{(x-1)^3}dx+4 \int \frac {e^{-e^{-x^2} \left (3 x+5 e^{x^2}\right )}}{(x-1)^2}dx+12 \int \frac {e^{-e^{-x^2} \left (e^{x^2} x^2+3 x+5 e^{x^2}\right )}}{(x-1)^2}dx+\int \frac {e^{x-e^{-x^2} \left (3 x+5 e^{x^2}\right )}}{(x-1)^2}dx+3 \int \frac {e^{x-e^{-x^2} \left (e^{x^2} x^2+3 x+5 e^{x^2}\right )}}{(x-1)^2}dx+36 \int \frac {e^{-e^{-x^2} \left (e^{x^2} x^2+3 x+5 e^{x^2}\right )}}{x-1}dx+12 \int \frac {e^{x-e^{-x^2} \left (e^{x^2} x^2+3 x+5 e^{x^2}\right )}}{x-1}dx-4 \int \frac {e^{-e^{-x^2} \left (3 x+5 e^{x^2}\right )}}{x^2}dx-12 \int \frac {e^{-e^{-x^2} \left (e^{x^2} x^2+3 x+5 e^{x^2}\right )}}{x}dx\) |
Int[(E^((E^x^2*(-5 + x) - 3*x)/E^x^2 - x - x^2)*(12*x - 12*x^2 - 24*x^3 + 24*x^4 + E^x*(3*x^2 - 3*x^3 - 6*x^4 + 6*x^5) + E^x^2*(4 - 12*x + E^x*(-3*x ^2 + x^3))))/(-x^2 + 3*x^3 - 3*x^4 + x^5),x]
3.28.48.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 2.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79
| method | result | size |
| risch | \(\frac {\left ({\mathrm e}^{x} x +4\right ) {\mathrm e}^{{\mathrm e}^{x^{2}} {\mathrm e}^{-x^{2}} x -5 \,{\mathrm e}^{x^{2}} {\mathrm e}^{-x^{2}}-3 x \,{\mathrm e}^{-x^{2}}-x}}{\left (x^{2}-2 x +1\right ) x}\) | \(59\) |
| parallelrisch | \(-\frac {\left (-2 x \,{\mathrm e}^{x} {\mathrm e}^{\left (\left (-5+x \right ) {\mathrm e}^{x^{2}}-3 x \right ) {\mathrm e}^{-x^{2}}}-8 \,{\mathrm e}^{\left (\left (-5+x \right ) {\mathrm e}^{x^{2}}-3 x \right ) {\mathrm e}^{-x^{2}}}\right ) {\mathrm e}^{-x}}{2 x \left (x^{2}-2 x +1\right )}\) | \(68\) |
int((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)*exp(x) +24*x^4-24*x^3-12*x^2+12*x)*exp(((-5+x)*exp(x^2)-3*x)/exp(x^2))/(x^5-3*x^4 +3*x^3-x^2)/exp(x)/exp(x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {e^{e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2} \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {{\left (x e^{x} + 4\right )} e^{\left (x^{2} - {\left ({\left (x^{2} + 5\right )} e^{\left (x^{2}\right )} + 3 \, x\right )} e^{\left (-x^{2}\right )}\right )}}{x^{3} - 2 \, x^{2} + x} \]
integrate((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)* exp(x)+24*x^4-24*x^3-12*x^2+12*x)*exp(((-5+x)*exp(x^2)-3*x)/exp(x^2))/(x^5 -3*x^4+3*x^3-x^2)/exp(x)/exp(x^2),x, algorithm=\
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {e^{e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2} \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {\left (x e^{x} + 4\right ) e^{\left (- 3 x + \left (x - 5\right ) e^{x^{2}}\right ) e^{- x^{2}}}}{x^{3} e^{x} - 2 x^{2} e^{x} + x e^{x}} \]
integrate((((x**3-3*x**2)*exp(x)-12*x+4)*exp(x**2)+(6*x**5-6*x**4-3*x**3+3 *x**2)*exp(x)+24*x**4-24*x**3-12*x**2+12*x)*exp(((-5+x)*exp(x**2)-3*x)/exp (x**2))/(x**5-3*x**4+3*x**3-x**2)/exp(x)/exp(x**2),x)
(x*exp(x) + 4)*exp((-3*x + (x - 5)*exp(x**2))*exp(-x**2))/(x**3*exp(x) - 2 *x**2*exp(x) + x*exp(x))
\[ \int \frac {e^{e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2} \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx=\int { \frac {{\left (24 \, x^{4} - 24 \, x^{3} - 12 \, x^{2} + {\left ({\left (x^{3} - 3 \, x^{2}\right )} e^{x} - 12 \, x + 4\right )} e^{\left (x^{2}\right )} + 3 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{3} + x^{2}\right )} e^{x} + 12 \, x\right )} e^{\left (-x^{2} + {\left ({\left (x - 5\right )} e^{\left (x^{2}\right )} - 3 \, x\right )} e^{\left (-x^{2}\right )} - x\right )}}{x^{5} - 3 \, x^{4} + 3 \, x^{3} - x^{2}} \,d x } \]
integrate((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)* exp(x)+24*x^4-24*x^3-12*x^2+12*x)*exp(((-5+x)*exp(x^2)-3*x)/exp(x^2))/(x^5 -3*x^4+3*x^3-x^2)/exp(x)/exp(x^2),x, algorithm=\
integrate((24*x^4 - 24*x^3 - 12*x^2 + ((x^3 - 3*x^2)*e^x - 12*x + 4)*e^(x^ 2) + 3*(2*x^5 - 2*x^4 - x^3 + x^2)*e^x + 12*x)*e^(-x^2 + ((x - 5)*e^(x^2) - 3*x)*e^(-x^2) - x)/(x^5 - 3*x^4 + 3*x^3 - x^2), x)
\[ \int \frac {e^{e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2} \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx=\int { \frac {{\left (24 \, x^{4} - 24 \, x^{3} - 12 \, x^{2} + {\left ({\left (x^{3} - 3 \, x^{2}\right )} e^{x} - 12 \, x + 4\right )} e^{\left (x^{2}\right )} + 3 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{3} + x^{2}\right )} e^{x} + 12 \, x\right )} e^{\left (-x^{2} + {\left ({\left (x - 5\right )} e^{\left (x^{2}\right )} - 3 \, x\right )} e^{\left (-x^{2}\right )} - x\right )}}{x^{5} - 3 \, x^{4} + 3 \, x^{3} - x^{2}} \,d x } \]
integrate((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)* exp(x)+24*x^4-24*x^3-12*x^2+12*x)*exp(((-5+x)*exp(x^2)-3*x)/exp(x^2))/(x^5 -3*x^4+3*x^3-x^2)/exp(x)/exp(x^2),x, algorithm=\
integrate((24*x^4 - 24*x^3 - 12*x^2 + ((x^3 - 3*x^2)*e^x - 12*x + 4)*e^(x^ 2) + 3*(2*x^5 - 2*x^4 - x^3 + x^2)*e^x + 12*x)*e^(-x^2 + ((x - 5)*e^(x^2) - 3*x)*e^(-x^2) - x)/(x^5 - 3*x^4 + 3*x^3 - x^2), x)
Timed out. \[ \int \frac {e^{e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2} \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx=\int -\frac {{\mathrm {e}}^{-{\mathrm {e}}^{-x^2}\,\left (3\,x-{\mathrm {e}}^{x^2}\,\left (x-5\right )\right )}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-x^2}\,\left (12\,x+{\mathrm {e}}^x\,\left (6\,x^5-6\,x^4-3\,x^3+3\,x^2\right )-12\,x^2-24\,x^3+24\,x^4-{\mathrm {e}}^{x^2}\,\left (12\,x+{\mathrm {e}}^x\,\left (3\,x^2-x^3\right )-4\right )\right )}{-x^5+3\,x^4-3\,x^3+x^2} \,d x \]
int(-(exp(-exp(-x^2)*(3*x - exp(x^2)*(x - 5)))*exp(-x)*exp(-x^2)*(12*x + e xp(x)*(3*x^2 - 3*x^3 - 6*x^4 + 6*x^5) - 12*x^2 - 24*x^3 + 24*x^4 - exp(x^2 )*(12*x + exp(x)*(3*x^2 - x^3) - 4)))/(x^2 - 3*x^3 + 3*x^4 - x^5),x)