Integrand size = 186, antiderivative size = 30 \[ \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx=e^x+\frac {3}{-8-\frac {16}{9} e^{2 x^2} \left (e^x-x\right )^2} \]
\[ \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx=\int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx \]
Integrate[(162*E^x + E^(4*x^2)*(8*E^(5*x) - 32*E^(4*x)*x + 48*E^(3*x)*x^2 - 32*E^(2*x)*x^3 + 8*E^x*x^4) + E^(2*x^2)*(72*E^(3*x) + E^(2*x)*(27 - 90*x ) + 27*x + 54*x^3 + E^x*(-27 - 27*x - 36*x^2)))/(162 + E^(2*x^2)*(72*E^(2* x) - 144*E^x*x + 72*x^2) + E^(4*x^2)*(8*E^(4*x) - 32*E^(3*x)*x + 48*E^(2*x )*x^2 - 32*E^x*x^3 + 8*x^4)),x]
Integrate[(162*E^x + E^(4*x^2)*(8*E^(5*x) - 32*E^(4*x)*x + 48*E^(3*x)*x^2 - 32*E^(2*x)*x^3 + 8*E^x*x^4) + E^(2*x^2)*(72*E^(3*x) + E^(2*x)*(27 - 90*x ) + 27*x + 54*x^3 + E^x*(-27 - 27*x - 36*x^2)))/(162 + E^(2*x^2)*(72*E^(2* x) - 144*E^x*x + 72*x^2) + E^(4*x^2)*(8*E^(4*x) - 32*E^(3*x)*x + 48*E^(2*x )*x^2 - 32*E^x*x^3 + 8*x^4)), 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.
| \(\displaystyle \int \frac {e^{2 x^2} \left (54 x^3+e^x \left (-36 x^2-27 x-27\right )+27 x+72 e^{3 x}+e^{2 x} (27-90 x)\right )+e^{4 x^2} \left (8 e^x x^4-32 e^{2 x} x^3+48 e^{3 x} x^2-32 e^{4 x} x+8 e^{5 x}\right )+162 e^x}{e^{2 x^2} \left (72 x^2-144 e^x x+72 e^{2 x}\right )+e^{4 x^2} \left (8 x^4-32 e^x x^3+48 e^{2 x} x^2-32 e^{3 x} x+8 e^{4 x}\right )+162} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 x^2} \left (54 x^3+e^x \left (-36 x^2-27 x-27\right )+27 x+72 e^{3 x}+e^{2 x} (27-90 x)\right )+e^{4 x^2} \left (8 e^x x^4-32 e^{2 x} x^3+48 e^{3 x} x^2-32 e^{4 x} x+8 e^{5 x}\right )+162 e^x}{2 \left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {8 e^{4 x^2} \left (e^x x^4-4 e^{2 x} x^3+6 e^{3 x} x^2-4 e^{4 x} x+e^{5 x}\right )+162 e^x+9 e^{2 x^2} \left (6 x^3+3 x+8 e^{3 x}+e^{2 x} (3-10 x)-e^x \left (4 x^2+3 x+3\right )\right )}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{4 x^2+x} x^4}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}+\frac {54 e^{2 x^2} x^3}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}-\frac {32 e^{2 x (2 x+1)} x^3}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}-\frac {36 e^{2 x^2+x} x^2}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}+\frac {48 e^{4 x^2+3 x} x^2}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}-\frac {27 e^{2 x^2+x} x}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}+\frac {27 e^{2 x^2} x}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}-\frac {90 e^{2 x (x+1)} x}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}-\frac {32 e^{4 x (x+1)} x}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}+\frac {162 e^x}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}-\frac {27 e^{2 x^2+x}}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}+\frac {27 e^{2 x (x+1)}}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}+\frac {72 e^{2 x^2+3 x}}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}+\frac {8 e^{4 x^2+5 x}}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (162 \int \frac {e^x}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}dx-27 \int \frac {e^{2 x^2+x}}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}dx-27 \int \frac {e^{2 x^2+x} x}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}dx-36 \int \frac {e^{2 x^2+x} x^2}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}dx+27 \int \frac {e^{2 x (x+1)}}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx+72 \int \frac {e^{2 x^2+3 x}}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx+8 \int \frac {e^{4 x^2+5 x}}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx+27 \int \frac {e^{2 x^2} x}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx-90 \int \frac {e^{2 x (x+1)} x}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx-32 \int \frac {e^{4 x (x+1)} x}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx+48 \int \frac {e^{4 x^2+3 x} x^2}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx+8 \int \frac {e^{4 x^2+x} x^4}{\left (-2 e^{2 x^2} x^2+4 e^{2 x^2+x} x-2 e^{2 x (x+1)}-9\right )^2}dx+54 \int \frac {e^{2 x^2} x^3}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx-32 \int \frac {e^{2 x (2 x+1)} x^3}{\left (2 e^{2 x^2} x^2-4 e^{2 x^2+x} x+2 e^{2 x (x+1)}+9\right )^2}dx\right )\) |
Int[(162*E^x + E^(4*x^2)*(8*E^(5*x) - 32*E^(4*x)*x + 48*E^(3*x)*x^2 - 32*E ^(2*x)*x^3 + 8*E^x*x^4) + E^(2*x^2)*(72*E^(3*x) + E^(2*x)*(27 - 90*x) + 27 *x + 54*x^3 + E^x*(-27 - 27*x - 36*x^2)))/(162 + E^(2*x^2)*(72*E^(2*x) - 1 44*E^x*x + 72*x^2) + E^(4*x^2)*(8*E^(4*x) - 32*E^(3*x)*x + 48*E^(2*x)*x^2 - 32*E^x*x^3 + 8*x^4)),x]
3.22.55.3.1 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]]
Time = 0.77 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \({\mathrm e}^{x}-\frac {27}{8 \left (2 \,{\mathrm e}^{2 \left (1+x \right ) x}-4 x \,{\mathrm e}^{\left (1+2 x \right ) x}+2 \,{\mathrm e}^{2 x^{2}} x^{2}+9\right )}\) | \(41\) |
| parallelrisch | \(\frac {16 \,{\mathrm e}^{2 x^{2}} {\mathrm e}^{x} x^{2}-27-32 \,{\mathrm e}^{2 x} {\mathrm e}^{2 x^{2}} x +16 \,{\mathrm e}^{3 x} {\mathrm e}^{2 x^{2}}+72 \,{\mathrm e}^{x}}{16 \,{\mathrm e}^{2 x} {\mathrm e}^{2 x^{2}}-32 \,{\mathrm e}^{2 x^{2}} {\mathrm e}^{x} x +16 \,{\mathrm e}^{2 x^{2}} x^{2}+72}\) | \(85\) |
int(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp(x)*x^ 4)*exp(x^2)^4+(72*exp(x)^3+(-90*x+27)*exp(x)^2+(-36*x^2-27*x-27)*exp(x)+54 *x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*exp(x)^2*x ^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2)*exp(x ^2)^2+162),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx=\frac {16 \, {\left (x^{2} e^{x} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 72 \, e^{x} - 27}{8 \, {\left (2 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 9\right )}} \]
integrate(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp (x)*x^4)*exp(x^2)^4+(72*exp(x)^3+(-90*x+27)*exp(x)^2+(-36*x^2-27*x-27)*exp (x)+54*x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*exp( x)^2*x^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2) *exp(x^2)^2+162),x, algorithm=\
1/8*(16*(x^2*e^x - 2*x*e^(2*x) + e^(3*x))*e^(2*x^2) + 72*e^x - 27)/(2*(x^2 - 2*x*e^x + e^(2*x))*e^(2*x^2) + 9)
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx=e^{x} - \frac {27}{\left (16 x^{2} - 32 x e^{x} + 16 e^{2 x}\right ) e^{2 x^{2}} + 72} \]
integrate(((8*exp(x)**5-32*x*exp(x)**4+48*x**2*exp(x)**3-32*exp(x)**2*x**3 +8*exp(x)*x**4)*exp(x**2)**4+(72*exp(x)**3+(-90*x+27)*exp(x)**2+(-36*x**2- 27*x-27)*exp(x)+54*x**3+27*x)*exp(x**2)**2+162*exp(x))/((8*exp(x)**4-32*x* exp(x)**3+48*exp(x)**2*x**2-32*exp(x)*x**3+8*x**4)*exp(x**2)**4+(72*exp(x) **2-144*exp(x)*x+72*x**2)*exp(x**2)**2+162),x)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx=\frac {16 \, {\left (x^{2} e^{x} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 72 \, e^{x} - 27}{8 \, {\left (2 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} + 9\right )}} \]
integrate(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp (x)*x^4)*exp(x^2)^4+(72*exp(x)^3+(-90*x+27)*exp(x)^2+(-36*x^2-27*x-27)*exp (x)+54*x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*exp( x)^2*x^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2) *exp(x^2)^2+162),x, algorithm=\
1/8*(16*(x^2*e^x - 2*x*e^(2*x) + e^(3*x))*e^(2*x^2) + 72*e^x - 27)/(2*(x^2 - 2*x*e^x + e^(2*x))*e^(2*x^2) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 3224 vs. \(2 (25) = 50\).
Time = 1.00 (sec) , antiderivative size = 3224, normalized size of antiderivative = 107.47 \[ \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx=\text {Too large to display} \]
integrate(((8*exp(x)^5-32*x*exp(x)^4+48*x^2*exp(x)^3-32*exp(x)^2*x^3+8*exp (x)*x^4)*exp(x^2)^4+(72*exp(x)^3+(-90*x+27)*exp(x)^2+(-36*x^2-27*x-27)*exp (x)+54*x^3+27*x)*exp(x^2)^2+162*exp(x))/((8*exp(x)^4-32*x*exp(x)^3+48*exp( x)^2*x^2-32*exp(x)*x^3+8*x^4)*exp(x^2)^4+(72*exp(x)^2-144*exp(x)*x+72*x^2) *exp(x^2)^2+162),x, algorithm=\
1/4*(64*x^8*e^(16*x^2 + x) + 2304*x^8*e^(14*x^2 + x) + 20736*x^8*e^(12*x^2 + x) - 256*x^7*e^(16*x^2 + 2*x) - 256*x^7*e^(16*x^2 + x) - 9216*x^7*e^(14 *x^2 + 2*x) - 2304*x^7*e^(14*x^2 + x) - 82944*x^7*e^(12*x^2 + 2*x) + 41472 *x^7*e^(12*x^2 + x) - 216*x^6*e^(14*x^2) - 7776*x^6*e^(12*x^2) - 69984*x^6 *e^(10*x^2) + 384*x^6*e^(16*x^2 + 3*x) + 1024*x^6*e^(16*x^2 + 2*x) + 384*x ^6*e^(16*x^2 + x) + 13824*x^6*e^(14*x^2 + 3*x) + 9216*x^6*e^(14*x^2 + 2*x) - 1152*x^6*e^(14*x^2 + x) + 124416*x^6*e^(12*x^2 + 3*x) - 165888*x^6*e^(1 2*x^2 + 2*x) + 51840*x^6*e^(12*x^2 + x) + 186624*x^6*e^(10*x^2 + x) + 864* x^5*e^(14*x^2) + 7776*x^5*e^(12*x^2) - 139968*x^5*e^(10*x^2) - 256*x^5*e^( 16*x^2 + 4*x) - 1536*x^5*e^(16*x^2 + 3*x) - 1536*x^5*e^(16*x^2 + 2*x) - 25 6*x^5*e^(16*x^2 + x) - 9216*x^5*e^(14*x^2 + 4*x) - 13824*x^5*e^(14*x^2 + 3 *x) + 5760*x^5*e^(14*x^2 + 2*x) - 720*x^5*e^(14*x^2 + x) - 82944*x^5*e^(12 *x^2 + 4*x) + 248832*x^5*e^(12*x^2 + 3*x) - 165888*x^5*e^(12*x^2 + 2*x) + 5184*x^5*e^(12*x^2 + x) - 373248*x^5*e^(10*x^2 + 2*x) + 513216*x^5*e^(10*x ^2 + x) - 1296*x^4*e^(14*x^2) + 4860*x^4*e^(12*x^2) - 139968*x^4*e^(10*x^2 ) - 314928*x^4*e^(8*x^2) + 64*x^4*e^(16*x^2 + 5*x) + 1024*x^4*e^(16*x^2 + 4*x) + 2304*x^4*e^(16*x^2 + 3*x) + 1024*x^4*e^(16*x^2 + 2*x) + 64*x^4*e^(1 6*x^2 + x) + 2304*x^4*e^(14*x^2 + 5*x) + 9216*x^4*e^(14*x^2 + 4*x) - 9792* x^4*e^(14*x^2 + 3*x) - 216*x^4*e^(14*x^2 + 2*x) + 2304*x^4*e^(14*x^2 + x) + 20736*x^4*e^(12*x^2 + 5*x) - 165888*x^4*e^(12*x^2 + 4*x) + 207360*x^4...
Timed out. \[ \int \frac {162 e^x+e^{4 x^2} \left (8 e^{5 x}-32 e^{4 x} x+48 e^{3 x} x^2-32 e^{2 x} x^3+8 e^x x^4\right )+e^{2 x^2} \left (72 e^{3 x}+e^{2 x} (27-90 x)+27 x+54 x^3+e^x \left (-27-27 x-36 x^2\right )\right )}{162+e^{2 x^2} \left (72 e^{2 x}-144 e^x x+72 x^2\right )+e^{4 x^2} \left (8 e^{4 x}-32 e^{3 x} x+48 e^{2 x} x^2-32 e^x x^3+8 x^4\right )} \, dx=\int \frac {162\,{\mathrm {e}}^x+{\mathrm {e}}^{4\,x^2}\,\left (8\,{\mathrm {e}}^{5\,x}-32\,x\,{\mathrm {e}}^{4\,x}+8\,x^4\,{\mathrm {e}}^x+48\,x^2\,{\mathrm {e}}^{3\,x}-32\,x^3\,{\mathrm {e}}^{2\,x}\right )+{\mathrm {e}}^{2\,x^2}\,\left (27\,x+72\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x\,\left (36\,x^2+27\,x+27\right )-{\mathrm {e}}^{2\,x}\,\left (90\,x-27\right )+54\,x^3\right )}{{\mathrm {e}}^{4\,x^2}\,\left (8\,{\mathrm {e}}^{4\,x}-32\,x\,{\mathrm {e}}^{3\,x}-32\,x^3\,{\mathrm {e}}^x+48\,x^2\,{\mathrm {e}}^{2\,x}+8\,x^4\right )+{\mathrm {e}}^{2\,x^2}\,\left (72\,{\mathrm {e}}^{2\,x}-144\,x\,{\mathrm {e}}^x+72\,x^2\right )+162} \,d x \]
int((162*exp(x) + exp(4*x^2)*(8*exp(5*x) - 32*x*exp(4*x) + 8*x^4*exp(x) + 48*x^2*exp(3*x) - 32*x^3*exp(2*x)) + exp(2*x^2)*(27*x + 72*exp(3*x) - exp( x)*(27*x + 36*x^2 + 27) - exp(2*x)*(90*x - 27) + 54*x^3))/(exp(4*x^2)*(8*e xp(4*x) - 32*x*exp(3*x) - 32*x^3*exp(x) + 48*x^2*exp(2*x) + 8*x^4) + exp(2 *x^2)*(72*exp(2*x) - 144*x*exp(x) + 72*x^2) + 162),x)
int((162*exp(x) + exp(4*x^2)*(8*exp(5*x) - 32*x*exp(4*x) + 8*x^4*exp(x) + 48*x^2*exp(3*x) - 32*x^3*exp(2*x)) + exp(2*x^2)*(27*x + 72*exp(3*x) - exp( x)*(27*x + 36*x^2 + 27) - exp(2*x)*(90*x - 27) + 54*x^3))/(exp(4*x^2)*(8*e xp(4*x) - 32*x*exp(3*x) - 32*x^3*exp(x) + 48*x^2*exp(2*x) + 8*x^4) + exp(2 *x^2)*(72*exp(2*x) - 144*x*exp(x) + 72*x^2) + 162), x)