Integrand size = 246, antiderivative size = 21 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=\left (-1+\frac {x}{x+\left (e^{x (4+x)}+x\right )^2}\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(21)=42\).
Time = 3.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.62 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {x \left (2 e^{2 x (4+x)}+x+4 e^{x (4+x)} x+2 x^2\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \]
Integrate[(2*x^4 + E^(16*x + 4*x^2)*(-2 + 16*x + 8*x^2) + E^(12*x + 3*x^2) *(-4*x + 48*x^2 + 24*x^3) + E^(8*x + 2*x^2)*(48*x^3 + 24*x^4) + E^(4*x + x ^2)*(4*x^3 + 16*x^4 + 8*x^5))/(E^(24*x + 6*x^2) + 6*E^(20*x + 5*x^2)*x + x ^3 + 3*x^4 + 3*x^5 + x^6 + E^(16*x + 4*x^2)*(3*x + 15*x^2) + E^(12*x + 3*x ^2)*(12*x^2 + 20*x^3) + E^(8*x + 2*x^2)*(3*x^2 + 18*x^3 + 15*x^4) + E^(4*x + x^2)*(6*x^3 + 12*x^4 + 6*x^5)),x]
-((x*(2*E^(2*x*(4 + x)) + x + 4*E^(x*(4 + x))*x + 2*x^2))/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^2)
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.
| \(\displaystyle \int \frac {2 x^4+e^{4 x^2+16 x} \left (8 x^2+16 x-2\right )+e^{3 x^2+12 x} \left (24 x^3+48 x^2-4 x\right )+e^{2 x^2+8 x} \left (24 x^4+48 x^3\right )+e^{x^2+4 x} \left (8 x^5+16 x^4+4 x^3\right )}{x^6+3 x^5+3 x^4+x^3+6 e^{5 x^2+20 x} x+e^{6 x^2+24 x}+e^{4 x^2+16 x} \left (15 x^2+3 x\right )+e^{3 x^2+12 x} \left (20 x^3+12 x^2\right )+e^{2 x^2+8 x} \left (15 x^4+18 x^3+3 x^2\right )+e^{x^2+4 x} \left (6 x^5+12 x^4+6 x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x+e^{x (x+4)}\right )^3 \left (e^{x (x+4)} \left (4 x^2+8 x-1\right )+x\right )}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (x+e^{x (x+4)}\right )^3 \left (x-e^{x (x+4)} \left (-4 x^2-8 x+1\right )\right )}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {\left (4 x^3+4 e^{x (x+4)} x^2+12 x^2+8 e^{x (x+4)} x+6 x-2 e^{x (x+4)}-1\right ) x^2}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}-\frac {2 \left (2 x^3+2 e^{x (x+4)} x^2+8 x^2+4 e^{x (x+4)} x+7 x-e^{x (x+4)}-1\right ) x}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}+\frac {4 x^2+8 x-1}{x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (2 \int \frac {e^{x^2+4 x} x^2}{\left (-x^2-2 e^{x (x+4)} x-x-e^{2 x (x+4)}\right )^3}dx-\int \frac {x^2}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx+2 \int \frac {x}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}dx+2 \int \frac {e^{x^2+4 x} x}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}dx-14 \int \frac {x^2}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}dx-8 \int \frac {e^{x^2+4 x} x^2}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}dx-\int \frac {1}{x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}}dx+8 \int \frac {x}{x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}}dx+4 \int \frac {x^2}{x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}}dx+4 \int \frac {x^5}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx+12 \int \frac {x^4}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx+4 \int \frac {e^{x^2+4 x} x^4}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx-4 \int \frac {x^4}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}dx+6 \int \frac {x^3}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx+8 \int \frac {e^{x^2+4 x} x^3}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^3}dx-16 \int \frac {x^3}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}dx-4 \int \frac {e^{x^2+4 x} x^3}{\left (x^2+2 e^{x (x+4)} x+x+e^{2 x (x+4)}\right )^2}dx\right )\) |
Int[(2*x^4 + E^(16*x + 4*x^2)*(-2 + 16*x + 8*x^2) + E^(12*x + 3*x^2)*(-4*x + 48*x^2 + 24*x^3) + E^(8*x + 2*x^2)*(48*x^3 + 24*x^4) + E^(4*x + x^2)*(4 *x^3 + 16*x^4 + 8*x^5))/(E^(24*x + 6*x^2) + 6*E^(20*x + 5*x^2)*x + x^3 + 3 *x^4 + 3*x^5 + x^6 + E^(16*x + 4*x^2)*(3*x + 15*x^2) + E^(12*x + 3*x^2)*(1 2*x^2 + 20*x^3) + E^(8*x + 2*x^2)*(3*x^2 + 18*x^3 + 15*x^4) + E^(4*x + x^2 )*(6*x^3 + 12*x^4 + 6*x^5)),x]
3.23.14.3.1 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. \(51\) vs. \(2(20)=40\).
Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.48
| method | result | size |
| risch | \(-\frac {\left (2 x^{2}+4 x \,{\mathrm e}^{\left (4+x \right ) x}+2 \,{\mathrm e}^{2 \left (4+x \right ) x}+x \right ) x}{\left ({\mathrm e}^{2 \left (4+x \right ) x}+2 x \,{\mathrm e}^{\left (4+x \right ) x}+x^{2}+x \right )^{2}}\) | \(52\) |
| parallelrisch | \(\frac {-2 x^{3}-4 x^{2} {\mathrm e}^{x^{2}+4 x}-2 \,{\mathrm e}^{2 x^{2}+8 x} x -x^{2}}{x^{4}+4 \,{\mathrm e}^{x^{2}+4 x} x^{3}+6 \,{\mathrm e}^{2 x^{2}+8 x} x^{2}+4 \,{\mathrm e}^{3 x^{2}+12 x} x +{\mathrm e}^{4 x^{2}+16 x}+2 x^{3}+4 x^{2} {\mathrm e}^{x^{2}+4 x}+2 \,{\mathrm e}^{2 x^{2}+8 x} x +x^{2}}\) | \(130\) |
int(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^3+(24* x^4+48*x^3)*exp(x^2+4*x)^2+(8*x^5+16*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/(exp(x ^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+12*x^2)*e xp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x^3)*ex p(x^2+4*x)+x^6+3*x^5+3*x^4+x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 5.24 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}}{x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (x^{2} + 4 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \]
integrate(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^ 3+(24*x^4+48*x^3)*exp(x^2+4*x)^2+(8*x^5+16*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/ (exp(x^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+12* x^2)*exp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x ^3)*exp(x^2+4*x)+x^6+3*x^5+3*x^4+x^3),x, algorithm=\
-(2*x^3 + 4*x^2*e^(x^2 + 4*x) + x^2 + 2*x*e^(2*x^2 + 8*x))/(x^4 + 2*x^3 + x^2 + 4*x*e^(3*x^2 + 12*x) + 2*(3*x^2 + x)*e^(2*x^2 + 8*x) + 4*(x^3 + x^2) *e^(x^2 + 4*x) + e^(4*x^2 + 16*x))
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (15) = 30\).
Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 5.19 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=\frac {- 2 x^{3} - 4 x^{2} e^{x^{2} + 4 x} - x^{2} - 2 x e^{2 x^{2} + 8 x}}{x^{4} + 2 x^{3} + x^{2} + 4 x e^{3 x^{2} + 12 x} + \left (6 x^{2} + 2 x\right ) e^{2 x^{2} + 8 x} + \left (4 x^{3} + 4 x^{2}\right ) e^{x^{2} + 4 x} + e^{4 x^{2} + 16 x}} \]
integrate(((8*x**2+16*x-2)*exp(x**2+4*x)**4+(24*x**3+48*x**2-4*x)*exp(x**2 +4*x)**3+(24*x**4+48*x**3)*exp(x**2+4*x)**2+(8*x**5+16*x**4+4*x**3)*exp(x* *2+4*x)+2*x**4)/(exp(x**2+4*x)**6+6*x*exp(x**2+4*x)**5+(15*x**2+3*x)*exp(x **2+4*x)**4+(20*x**3+12*x**2)*exp(x**2+4*x)**3+(15*x**4+18*x**3+3*x**2)*ex p(x**2+4*x)**2+(6*x**5+12*x**4+6*x**3)*exp(x**2+4*x)+x**6+3*x**5+3*x**4+x* *3),x)
(-2*x**3 - 4*x**2*exp(x**2 + 4*x) - x**2 - 2*x*exp(2*x**2 + 8*x))/(x**4 + 2*x**3 + x**2 + 4*x*exp(3*x**2 + 12*x) + (6*x**2 + 2*x)*exp(2*x**2 + 8*x) + (4*x**3 + 4*x**2)*exp(x**2 + 4*x) + exp(4*x**2 + 16*x))
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 5.24 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}}{x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (x^{2} + 4 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \]
integrate(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^ 3+(24*x^4+48*x^3)*exp(x^2+4*x)^2+(8*x^5+16*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/ (exp(x^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+12* x^2)*exp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x ^3)*exp(x^2+4*x)+x^6+3*x^5+3*x^4+x^3),x, algorithm=\
-(2*x^3 + 4*x^2*e^(x^2 + 4*x) + x^2 + 2*x*e^(2*x^2 + 8*x))/(x^4 + 2*x^3 + x^2 + 4*x*e^(3*x^2 + 12*x) + 2*(3*x^2 + x)*e^(2*x^2 + 8*x) + 4*(x^3 + x^2) *e^(x^2 + 4*x) + e^(4*x^2 + 16*x))
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (20) = 40\).
Time = 0.66 (sec) , antiderivative size = 128, normalized size of antiderivative = 6.10 \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=-\frac {2 \, {\left (2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}\right )}}{x^{4} + 4 \, x^{3} e^{\left (x^{2} + 4 \, x\right )} + 2 \, x^{3} + 6 \, x^{2} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \]
integrate(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^ 3+(24*x^4+48*x^3)*exp(x^2+4*x)^2+(8*x^5+16*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/ (exp(x^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+12* x^2)*exp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x ^3)*exp(x^2+4*x)+x^6+3*x^5+3*x^4+x^3),x, algorithm=\
-2*(2*x^3 + 4*x^2*e^(x^2 + 4*x) + x^2 + 2*x*e^(2*x^2 + 8*x))/(x^4 + 4*x^3* e^(x^2 + 4*x) + 2*x^3 + 6*x^2*e^(2*x^2 + 8*x) + 4*x^2*e^(x^2 + 4*x) + x^2 + 4*x*e^(3*x^2 + 12*x) + 2*x*e^(2*x^2 + 8*x) + e^(4*x^2 + 16*x))
Timed out. \[ \int \frac {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx=\int \frac {{\mathrm {e}}^{4\,x^2+16\,x}\,\left (8\,x^2+16\,x-2\right )+{\mathrm {e}}^{3\,x^2+12\,x}\,\left (24\,x^3+48\,x^2-4\,x\right )+{\mathrm {e}}^{x^2+4\,x}\,\left (8\,x^5+16\,x^4+4\,x^3\right )+2\,x^4+{\mathrm {e}}^{2\,x^2+8\,x}\,\left (24\,x^4+48\,x^3\right )}{{\mathrm {e}}^{6\,x^2+24\,x}+{\mathrm {e}}^{2\,x^2+8\,x}\,\left (15\,x^4+18\,x^3+3\,x^2\right )+{\mathrm {e}}^{4\,x^2+16\,x}\,\left (15\,x^2+3\,x\right )+6\,x\,{\mathrm {e}}^{5\,x^2+20\,x}+{\mathrm {e}}^{x^2+4\,x}\,\left (6\,x^5+12\,x^4+6\,x^3\right )+x^3+3\,x^4+3\,x^5+x^6+{\mathrm {e}}^{3\,x^2+12\,x}\,\left (20\,x^3+12\,x^2\right )} \,d x \]
int((exp(16*x + 4*x^2)*(16*x + 8*x^2 - 2) + exp(12*x + 3*x^2)*(48*x^2 - 4* x + 24*x^3) + exp(4*x + x^2)*(4*x^3 + 16*x^4 + 8*x^5) + 2*x^4 + exp(8*x + 2*x^2)*(48*x^3 + 24*x^4))/(exp(24*x + 6*x^2) + exp(8*x + 2*x^2)*(3*x^2 + 1 8*x^3 + 15*x^4) + exp(16*x + 4*x^2)*(3*x + 15*x^2) + 6*x*exp(20*x + 5*x^2) + exp(4*x + x^2)*(6*x^3 + 12*x^4 + 6*x^5) + x^3 + 3*x^4 + 3*x^5 + x^6 + e xp(12*x + 3*x^2)*(12*x^2 + 20*x^3)),x)
int((exp(16*x + 4*x^2)*(16*x + 8*x^2 - 2) + exp(12*x + 3*x^2)*(48*x^2 - 4* x + 24*x^3) + exp(4*x + x^2)*(4*x^3 + 16*x^4 + 8*x^5) + 2*x^4 + exp(8*x + 2*x^2)*(48*x^3 + 24*x^4))/(exp(24*x + 6*x^2) + exp(8*x + 2*x^2)*(3*x^2 + 1 8*x^3 + 15*x^4) + exp(16*x + 4*x^2)*(3*x + 15*x^2) + 6*x*exp(20*x + 5*x^2) + exp(4*x + x^2)*(6*x^3 + 12*x^4 + 6*x^5) + x^3 + 3*x^4 + 3*x^5 + x^6 + e xp(12*x + 3*x^2)*(12*x^2 + 20*x^3)), x)