Integrand size = 147, antiderivative size = 26 \[ \int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}} \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx=-16+e^{\frac {4 (1+x)^2}{\left (e^{x^2} (-1+x)-x\right )^2}} \]
Time = 1.56 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}} \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx=e^{\frac {4 (1+x)^2}{\left (-e^{x^2} (-1+x)+x\right )^2}} \]
Integrate[(E^((4 + 8*x + 4*x^2)/(x^2 + E^x^2*(2*x - 2*x^2) + E^(2*x^2)*(1 - 2*x + x^2)))*(8 + 8*x + E^x^2*(-16 + 16*x^2 - 16*x^3 - 16*x^4)))/(-x^3 + E^(2*x^2)*(-3*x + 6*x^2 - 3*x^3) + E^(3*x^2)*(-1 + 3*x - 3*x^2 + x^3) + E ^x^2*(-3*x^2 + 3*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.
| \(\displaystyle \int \frac {\left (e^{x^2} \left (-16 x^4-16 x^3+16 x^2-16\right )+8 x+8\right ) \exp \left (\frac {4 x^2+8 x+4}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (x^2-2 x+1\right )}\right )}{-x^3+e^{2 x^2} \left (-3 x^3+6 x^2-3 x\right )+e^{3 x^2} \left (x^3-3 x^2+3 x-1\right )+e^{x^2} \left (3 x^3-3 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8 e^{\frac {4 (x+1)^2}{\left (x-e^{x^2} (x-1)\right )^2}} (x+1) \left (1-2 e^{x^2} \left (x^3-x+1\right )\right )}{\left (e^{x^2} (x-1)-x\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \int -\frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} (x+1) \left (1-2 e^{x^2} \left (x^3-x+1\right )\right )}{\left (e^{x^2} (1-x)+x\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -8 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} (x+1) \left (1-2 e^{x^2} \left (x^3-x+1\right )\right )}{\left (e^{x^2} (1-x)+x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -8 \int \left (\frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} \left (2 x^3-2 x^2+1\right ) (x+1)^2}{(x-1) \left (e^{x^2} x-x-e^{x^2}\right )^3}+\frac {2 e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} \left (x^4+x^3-x^2+1\right )}{(x-1) \left (e^{x^2} x-x-e^{x^2}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -8 \left (3 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}}}{\left (e^{x^2} x-x-e^{x^2}\right )^3}dx+4 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}}}{(x-1) \left (e^{x^2} x-x-e^{x^2}\right )^3}dx+\int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} x}{\left (e^{x^2} x-x-e^{x^2}\right )^3}dx+2 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} x^2}{\left (e^{x^2} x-x-e^{x^2}\right )^3}dx+2 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}}}{\left (e^{x^2} x-x-e^{x^2}\right )^2}dx+4 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}}}{(x-1) \left (e^{x^2} x-x-e^{x^2}\right )^2}dx+2 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} x}{\left (e^{x^2} x-x-e^{x^2}\right )^2}dx+4 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} x^2}{\left (e^{x^2} x-x-e^{x^2}\right )^2}dx+2 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} x^4}{\left (e^{x^2} x-x-e^{x^2}\right )^3}dx+4 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} x^3}{\left (e^{x^2} x-x-e^{x^2}\right )^3}dx+2 \int \frac {e^{\frac {4 (x+1)^2}{\left (e^{x^2} (1-x)+x\right )^2}} x^3}{\left (e^{x^2} x-x-e^{x^2}\right )^2}dx\right )\) |
Int[(E^((4 + 8*x + 4*x^2)/(x^2 + E^x^2*(2*x - 2*x^2) + E^(2*x^2)*(1 - 2*x + x^2)))*(8 + 8*x + E^x^2*(-16 + 16*x^2 - 16*x^3 - 16*x^4)))/(-x^3 + E^(2* x^2)*(-3*x + 6*x^2 - 3*x^3) + E^(3*x^2)*(-1 + 3*x - 3*x^2 + x^3) + E^x^2*( -3*x^2 + 3*x^3)),x]
3.26.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 2.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15
| method | result | size |
| risch | \({\mathrm e}^{\frac {4 \left (1+x \right )^{2}}{-2 x^{2} {\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}} x^{2}+2 \,{\mathrm e}^{x^{2}} x -2 x \,{\mathrm e}^{2 x^{2}}+x^{2}+{\mathrm e}^{2 x^{2}}}}\) | \(56\) |
| parallelrisch | \({\mathrm e}^{\frac {4 x^{2}+8 x +4}{-2 x^{2} {\mathrm e}^{x^{2}}+{\mathrm e}^{2 x^{2}} x^{2}+2 \,{\mathrm e}^{x^{2}} x -2 x \,{\mathrm e}^{2 x^{2}}+x^{2}+{\mathrm e}^{2 x^{2}}}}\) | \(59\) |
int(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/((x^2-2* x+1)*exp(x^2)^2+(-2*x^2+2*x)*exp(x^2)+x^2))/((x^3-3*x^2+3*x-1)*exp(x^2)^3+ (-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x,method=_RETURN VERBOSE)
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}} \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx=e^{\left (\frac {4 \, {\left (x^{2} + 2 \, x + 1\right )}}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )}}\right )} \]
integrate(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/(( x^2-2*x+1)*exp(x^2)^2+(-2*x^2+2*x)*exp(x^2)+x^2))/((x^3-3*x^2+3*x-1)*exp(x ^2)^3+(-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x, algorit hm=\
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}} \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx=e^{\frac {4 x^{2} + 8 x + 4}{x^{2} + \left (- 2 x^{2} + 2 x\right ) e^{x^{2}} + \left (x^{2} - 2 x + 1\right ) e^{2 x^{2}}}} \]
integrate(((-16*x**4-16*x**3+16*x**2-16)*exp(x**2)+8*x+8)*exp((4*x**2+8*x+ 4)/((x**2-2*x+1)*exp(x**2)**2+(-2*x**2+2*x)*exp(x**2)+x**2))/((x**3-3*x**2 +3*x-1)*exp(x**2)**3+(-3*x**3+6*x**2-3*x)*exp(x**2)**2+(3*x**3-3*x**2)*exp (x**2)-x**3),x)
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (24) = 48\).
Time = 0.71 (sec) , antiderivative size = 267, normalized size of antiderivative = 10.27 \[ \int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}} \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx=e^{\left (\frac {4 \, e^{\left (2 \, x^{2}\right )}}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (4 \, x^{2}\right )} - 2 \, {\left (2 \, x^{2} - 3 \, x + 1\right )} e^{\left (3 \, x^{2}\right )} + {\left (6 \, x^{2} - 6 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (2 \, x^{2} - x\right )} e^{\left (x^{2}\right )}} - \frac {8 \, e^{\left (x^{2}\right )}}{x^{2} - {\left (x^{2} - 2 \, x + 1\right )} e^{\left (3 \, x^{2}\right )} + {\left (3 \, x^{2} - 4 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - {\left (3 \, x^{2} - 2 \, x\right )} e^{\left (x^{2}\right )}} + \frac {8 \, e^{\left (x^{2}\right )}}{{\left (x - 1\right )} e^{\left (3 \, x^{2}\right )} - {\left (3 \, x - 2\right )} e^{\left (2 \, x^{2}\right )} + {\left (3 \, x - 1\right )} e^{\left (x^{2}\right )} - x} + \frac {4}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )}} + \frac {8}{{\left (x - 1\right )} e^{\left (2 \, x^{2}\right )} - {\left (2 \, x - 1\right )} e^{\left (x^{2}\right )} + x} + \frac {4}{e^{\left (2 \, x^{2}\right )} - 2 \, e^{\left (x^{2}\right )} + 1}\right )} \]
integrate(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/(( x^2-2*x+1)*exp(x^2)^2+(-2*x^2+2*x)*exp(x^2)+x^2))/((x^3-3*x^2+3*x-1)*exp(x ^2)^3+(-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x, algorit hm=\
e^(4*e^(2*x^2)/(x^2 + (x^2 - 2*x + 1)*e^(4*x^2) - 2*(2*x^2 - 3*x + 1)*e^(3 *x^2) + (6*x^2 - 6*x + 1)*e^(2*x^2) - 2*(2*x^2 - x)*e^(x^2)) - 8*e^(x^2)/( x^2 - (x^2 - 2*x + 1)*e^(3*x^2) + (3*x^2 - 4*x + 1)*e^(2*x^2) - (3*x^2 - 2 *x)*e^(x^2)) + 8*e^(x^2)/((x - 1)*e^(3*x^2) - (3*x - 2)*e^(2*x^2) + (3*x - 1)*e^(x^2) - x) + 4/(x^2 + (x^2 - 2*x + 1)*e^(2*x^2) - 2*(x^2 - x)*e^(x^2 )) + 8/((x - 1)*e^(2*x^2) - (2*x - 1)*e^(x^2) + x) + 4/(e^(2*x^2) - 2*e^(x ^2) + 1))
\[ \int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}} \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx=\int { \frac {8 \, {\left (2 \, {\left (x^{4} + x^{3} - x^{2} + 1\right )} e^{\left (x^{2}\right )} - x - 1\right )} e^{\left (\frac {4 \, {\left (x^{2} + 2 \, x + 1\right )}}{x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )}}\right )}}{x^{3} - {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} e^{\left (3 \, x^{2}\right )} + 3 \, {\left (x^{3} - 2 \, x^{2} + x\right )} e^{\left (2 \, x^{2}\right )} - 3 \, {\left (x^{3} - x^{2}\right )} e^{\left (x^{2}\right )}} \,d x } \]
integrate(((-16*x^4-16*x^3+16*x^2-16)*exp(x^2)+8*x+8)*exp((4*x^2+8*x+4)/(( x^2-2*x+1)*exp(x^2)^2+(-2*x^2+2*x)*exp(x^2)+x^2))/((x^3-3*x^2+3*x-1)*exp(x ^2)^3+(-3*x^3+6*x^2-3*x)*exp(x^2)^2+(3*x^3-3*x^2)*exp(x^2)-x^3),x, algorit hm=\
integrate(8*(2*(x^4 + x^3 - x^2 + 1)*e^(x^2) - x - 1)*e^(4*(x^2 + 2*x + 1) /(x^2 + (x^2 - 2*x + 1)*e^(2*x^2) - 2*(x^2 - x)*e^(x^2)))/(x^3 - (x^3 - 3* x^2 + 3*x - 1)*e^(3*x^2) + 3*(x^3 - 2*x^2 + x)*e^(2*x^2) - 3*(x^3 - x^2)*e ^(x^2)), x)
Time = 14.47 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.96 \[ \int \frac {e^{\frac {4+8 x+4 x^2}{x^2+e^{x^2} \left (2 x-2 x^2\right )+e^{2 x^2} \left (1-2 x+x^2\right )}} \left (8+8 x+e^{x^2} \left (-16+16 x^2-16 x^3-16 x^4\right )\right )}{-x^3+e^{2 x^2} \left (-3 x+6 x^2-3 x^3\right )+e^{3 x^2} \left (-1+3 x-3 x^2+x^3\right )+e^{x^2} \left (-3 x^2+3 x^3\right )} \, dx={\mathrm {e}}^{\frac {8\,x}{{\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}-2\,x\,{\mathrm {e}}^{2\,x^2}-2\,x^2\,{\mathrm {e}}^{x^2}+x^2\,{\mathrm {e}}^{2\,x^2}+x^2}}\,{\mathrm {e}}^{\frac {4\,x^2}{{\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}-2\,x\,{\mathrm {e}}^{2\,x^2}-2\,x^2\,{\mathrm {e}}^{x^2}+x^2\,{\mathrm {e}}^{2\,x^2}+x^2}}\,{\mathrm {e}}^{\frac {4}{{\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}-2\,x\,{\mathrm {e}}^{2\,x^2}-2\,x^2\,{\mathrm {e}}^{x^2}+x^2\,{\mathrm {e}}^{2\,x^2}+x^2}} \]
int(-(exp((8*x + 4*x^2 + 4)/(exp(x^2)*(2*x - 2*x^2) + exp(2*x^2)*(x^2 - 2* x + 1) + x^2))*(8*x - exp(x^2)*(16*x^3 - 16*x^2 + 16*x^4 + 16) + 8))/(exp( x^2)*(3*x^2 - 3*x^3) - exp(3*x^2)*(3*x - 3*x^2 + x^3 - 1) + exp(2*x^2)*(3* x - 6*x^2 + 3*x^3) + x^3),x)