Integrand size = 179, antiderivative size = 34 \[ \int \frac {e^{-2+\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}} \left (18 x^8+6 x^9+e^{4 x} \left (-6+10 x+4 x^2\right )+e^2 \left (-36 x^3-9 x^4\right )+e^{3 x} \left (36 x^3+12 x^4\right )+e^{2 x} \left (36 x^4+48 x^5+12 x^6\right )+e^x \left (48 x^6+28 x^7+4 x^8\right )\right )}{243 x^3+162 x^4+27 x^5} \, dx=\frac {e^{-x+\frac {\left (e^x+x^2\right )^4}{9 e^2 x^2}}}{3 (3+x)} \]
Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int \frac {e^{-2+\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}} \left (18 x^8+6 x^9+e^{4 x} \left (-6+10 x+4 x^2\right )+e^2 \left (-36 x^3-9 x^4\right )+e^{3 x} \left (36 x^3+12 x^4\right )+e^{2 x} \left (36 x^4+48 x^5+12 x^6\right )+e^x \left (48 x^6+28 x^7+4 x^8\right )\right )}{243 x^3+162 x^4+27 x^5} \, dx=\frac {e^{\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}}}{3 (3+x)} \]
Integrate[(E^(-2 + (E^(4*x) + 4*E^(3*x)*x^2 - 9*E^2*x^3 + 6*E^(2*x)*x^4 + 4*E^x*x^6 + x^8)/(9*E^2*x^2))*(18*x^8 + 6*x^9 + E^(4*x)*(-6 + 10*x + 4*x^2 ) + E^2*(-36*x^3 - 9*x^4) + E^(3*x)*(36*x^3 + 12*x^4) + E^(2*x)*(36*x^4 + 48*x^5 + 12*x^6) + E^x*(48*x^6 + 28*x^7 + 4*x^8)))/(243*x^3 + 162*x^4 + 27 *x^5),x]
E^((E^(4*x) + 4*E^(3*x)*x^2 - 9*E^2*x^3 + 6*E^(2*x)*x^4 + 4*E^x*x^6 + x^8) /(9*E^2*x^2))/(3*(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.
| \(\displaystyle \int \frac {\left (6 x^9+18 x^8+e^{4 x} \left (4 x^2+10 x-6\right )+e^2 \left (-9 x^4-36 x^3\right )+e^{3 x} \left (12 x^4+36 x^3\right )+e^x \left (4 x^8+28 x^7+48 x^6\right )+e^{2 x} \left (12 x^6+48 x^5+36 x^4\right )\right ) \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2\right )}{27 x^5+162 x^4+243 x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (6 x^9+18 x^8+e^{4 x} \left (4 x^2+10 x-6\right )+e^2 \left (-9 x^4-36 x^3\right )+e^{3 x} \left (12 x^4+36 x^3\right )+e^x \left (4 x^8+28 x^7+48 x^6\right )+e^{2 x} \left (12 x^6+48 x^5+36 x^4\right )\right ) \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2\right )}{x^3 \left (27 x^2+162 x+243\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (6 x^9+18 x^8+e^{4 x} \left (4 x^2+10 x-6\right )+e^2 \left (-9 x^4-36 x^3\right )+e^{3 x} \left (12 x^4+36 x^3\right )+e^x \left (4 x^8+28 x^7+48 x^6\right )+e^{2 x} \left (12 x^6+48 x^5+36 x^4\right )\right ) \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2\right )}{x^3 \left (3 \sqrt {3} x+9 \sqrt {3}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 (x+4) x^3 \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}+x-2\right )}{27 (x+3)}+\frac {4 (x+1) x \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}+2 x-2\right )}{9 (x+3)}+\frac {4 \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}+3 x-2\right )}{9 (x+3)}+\frac {2 (2 x-1) \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}+4 x-2\right )}{27 (x+3) x^3}+\frac {\left (2 x^6+6 x^5-3 e^2 x-12 e^2\right ) \exp \left (\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2\right )}{9 (x+3)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{3} \int e^{-\frac {-x^8-4 e^x x^6-6 e^{2 x} x^4-4 e^{3 x} x^2+18 e^2 x^2-e^{4 x}}{9 e^2 x^2}}dx+18 \int e^{\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2}dx-\frac {8}{9} \int e^{2 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}dx-\frac {2}{81} \int \frac {e^{4 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}}{x^3}dx+\frac {14}{243} \int \frac {e^{4 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}}{x^2}dx-\frac {14}{729} \int \frac {e^{4 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}}{x}dx-\frac {4}{9} \int e^{-\frac {-x^8-4 e^x x^6-6 e^{2 x} x^4-4 e^{3 x} x^2+18 e^2 x^2-e^{4 x}}{9 e^2 x^2}} xdx-6 \int e^{\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2} xdx+\frac {4}{9} \int e^{2 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}} xdx+\frac {4}{27} \int e^{-\frac {-x^8-4 e^x x^6-6 e^{2 x} x^4-4 e^{3 x} x^2+18 e^2 x^2-e^{4 x}}{9 e^2 x^2}} x^2dx+2 \int e^{\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2} x^2dx+\frac {4}{27} \int e^{-\frac {-x^8-4 e^x x^6-6 e^{2 x} x^4-4 e^{3 x} x^2+18 e^2 x^2-e^{4 x}}{9 e^2 x^2}} x^3dx-\frac {2}{3} \int e^{\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2} x^3dx+\frac {2}{9} \int e^{\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2} x^4dx-\frac {1}{3} \int \frac {e^{\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}}{(x+3)^2}dx-4 \int \frac {e^{-\frac {-x^8-4 e^x x^6-6 e^{2 x} x^4-4 e^{3 x} x^2+18 e^2 x^2-e^{4 x}}{9 e^2 x^2}}}{x+3}dx-\frac {1}{3} \left (162+e^2\right ) \int \frac {e^{\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}-2}}{x+3}dx+\frac {8}{3} \int \frac {e^{2 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}}{x+3}dx+\frac {4}{9} \int \frac {e^{3 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}}{x+3}dx+\frac {14}{729} \int \frac {e^{4 x-2+\frac {x^8+4 e^x x^6+6 e^{2 x} x^4-9 e^2 x^3+4 e^{3 x} x^2+e^{4 x}}{9 e^2 x^2}}}{x+3}dx\) |
Int[(E^(-2 + (E^(4*x) + 4*E^(3*x)*x^2 - 9*E^2*x^3 + 6*E^(2*x)*x^4 + 4*E^x* x^6 + x^8)/(9*E^2*x^2))*(18*x^8 + 6*x^9 + E^(4*x)*(-6 + 10*x + 4*x^2) + E^ 2*(-36*x^3 - 9*x^4) + E^(3*x)*(36*x^3 + 12*x^4) + E^(2*x)*(36*x^4 + 48*x^5 + 12*x^6) + E^x*(48*x^6 + 28*x^7 + 4*x^8)))/(243*x^3 + 162*x^4 + 27*x^5), x]
3.14.86.3.1 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])
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(29)=58\).
Time = 11.62 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.76
| method | result | size |
| risch | \(\frac {{\mathrm e}^{-\frac {\left (-{\mathrm e}^{4 x}-4 x^{2} {\mathrm e}^{3 x}-6 \,{\mathrm e}^{2 x} x^{4}-4 x^{6} {\mathrm e}^{x}+9 x^{3} {\mathrm e}^{2}-x^{8}\right ) {\mathrm e}^{-2}}{9 x^{2}}}}{3 x +9}\) | \(60\) |
| parallelrisch | \(\frac {{\mathrm e}^{-\frac {\left (-{\mathrm e}^{4 x}-4 x^{2} {\mathrm e}^{3 x}-6 \,{\mathrm e}^{2 x} x^{4}-4 x^{6} {\mathrm e}^{x}+9 x^{3} {\mathrm e}^{2}-x^{8}\right ) {\mathrm e}^{-2}}{9 x^{2}}}}{3 x +9}\) | \(64\) |
int(((4*x^2+10*x-6)*exp(x)^4+(12*x^4+36*x^3)*exp(x)^3+(12*x^6+48*x^5+36*x^ 4)*exp(x)^2+(4*x^8+28*x^7+48*x^6)*exp(x)+(-9*x^4-36*x^3)*exp(1)^2+6*x^9+18 *x^8)*exp(1/9*(exp(x)^4+4*x^2*exp(x)^3+6*exp(x)^2*x^4+4*x^6*exp(x)-9*x^3*e xp(1)^2+x^8)/x^2/exp(1)^2)/(27*x^5+162*x^4+243*x^3)/exp(1)^2,x,method=_RET URNVERBOSE)
1/3/(3+x)*exp(-1/9*(-exp(4*x)-4*x^2*exp(3*x)-6*exp(2*x)*x^4-4*x^6*exp(x)+9 *x^3*exp(2)-x^8)/x^2*exp(-2))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.85 \[ \int \frac {e^{-2+\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}} \left (18 x^8+6 x^9+e^{4 x} \left (-6+10 x+4 x^2\right )+e^2 \left (-36 x^3-9 x^4\right )+e^{3 x} \left (36 x^3+12 x^4\right )+e^{2 x} \left (36 x^4+48 x^5+12 x^6\right )+e^x \left (48 x^6+28 x^7+4 x^8\right )\right )}{243 x^3+162 x^4+27 x^5} \, dx=\frac {e^{\left (\frac {{\left (x^{8} + 4 \, x^{6} e^{x} + 6 \, x^{4} e^{\left (2 \, x\right )} + 4 \, x^{2} e^{\left (3 \, x\right )} - 9 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{2} + e^{\left (4 \, x\right )}\right )} e^{\left (-2\right )}}{9 \, x^{2}} + 2\right )}}{3 \, {\left (x + 3\right )}} \]
integrate(((4*x^2+10*x-6)*exp(x)^4+(12*x^4+36*x^3)*exp(x)^3+(12*x^6+48*x^5 +36*x^4)*exp(x)^2+(4*x^8+28*x^7+48*x^6)*exp(x)+(-9*x^4-36*x^3)*exp(1)^2+6* x^9+18*x^8)*exp(1/9*(exp(x)^4+4*x^2*exp(x)^3+6*exp(x)^2*x^4+4*x^6*exp(x)-9 *x^3*exp(1)^2+x^8)/x^2/exp(1)^2)/(27*x^5+162*x^4+243*x^3)/exp(1)^2,x, algo rithm=\
1/3*e^(1/9*(x^8 + 4*x^6*e^x + 6*x^4*e^(2*x) + 4*x^2*e^(3*x) - 9*(x^3 + 2*x ^2)*e^2 + e^(4*x))*e^(-2)/x^2 + 2)/(x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {e^{-2+\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}} \left (18 x^8+6 x^9+e^{4 x} \left (-6+10 x+4 x^2\right )+e^2 \left (-36 x^3-9 x^4\right )+e^{3 x} \left (36 x^3+12 x^4\right )+e^{2 x} \left (36 x^4+48 x^5+12 x^6\right )+e^x \left (48 x^6+28 x^7+4 x^8\right )\right )}{243 x^3+162 x^4+27 x^5} \, dx=\frac {e^{\frac {\frac {x^{8}}{9} + \frac {4 x^{6} e^{x}}{9} + \frac {2 x^{4} e^{2 x}}{3} - x^{3} e^{2} + \frac {4 x^{2} e^{3 x}}{9} + \frac {e^{4 x}}{9}}{x^{2} e^{2}}}}{3 x + 9} \]
integrate(((4*x**2+10*x-6)*exp(x)**4+(12*x**4+36*x**3)*exp(x)**3+(12*x**6+ 48*x**5+36*x**4)*exp(x)**2+(4*x**8+28*x**7+48*x**6)*exp(x)+(-9*x**4-36*x** 3)*exp(1)**2+6*x**9+18*x**8)*exp(1/9*(exp(x)**4+4*x**2*exp(x)**3+6*exp(x)* *2*x**4+4*x**6*exp(x)-9*x**3*exp(1)**2+x**8)/x**2/exp(1)**2)/(27*x**5+162* x**4+243*x**3)/exp(1)**2,x)
exp((x**8/9 + 4*x**6*exp(x)/9 + 2*x**4*exp(2*x)/3 - x**3*exp(2) + 4*x**2*e xp(3*x)/9 + exp(4*x)/9)*exp(-2)/x**2)/(3*x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
Time = 0.55 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71 \[ \int \frac {e^{-2+\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}} \left (18 x^8+6 x^9+e^{4 x} \left (-6+10 x+4 x^2\right )+e^2 \left (-36 x^3-9 x^4\right )+e^{3 x} \left (36 x^3+12 x^4\right )+e^{2 x} \left (36 x^4+48 x^5+12 x^6\right )+e^x \left (48 x^6+28 x^7+4 x^8\right )\right )}{243 x^3+162 x^4+27 x^5} \, dx=\frac {e^{\left (\frac {1}{9} \, x^{6} e^{\left (-2\right )} + \frac {4}{9} \, x^{4} e^{\left (x - 2\right )} + \frac {2}{3} \, x^{2} e^{\left (2 \, x - 2\right )} - x + \frac {e^{\left (4 \, x - 2\right )}}{9 \, x^{2}} + \frac {4}{9} \, e^{\left (3 \, x - 2\right )}\right )}}{3 \, {\left (x + 3\right )}} \]
integrate(((4*x^2+10*x-6)*exp(x)^4+(12*x^4+36*x^3)*exp(x)^3+(12*x^6+48*x^5 +36*x^4)*exp(x)^2+(4*x^8+28*x^7+48*x^6)*exp(x)+(-9*x^4-36*x^3)*exp(1)^2+6* x^9+18*x^8)*exp(1/9*(exp(x)^4+4*x^2*exp(x)^3+6*exp(x)^2*x^4+4*x^6*exp(x)-9 *x^3*exp(1)^2+x^8)/x^2/exp(1)^2)/(27*x^5+162*x^4+243*x^3)/exp(1)^2,x, algo rithm=\
1/3*e^(1/9*x^6*e^(-2) + 4/9*x^4*e^(x - 2) + 2/3*x^2*e^(2*x - 2) - x + 1/9* e^(4*x - 2)/x^2 + 4/9*e^(3*x - 2))/(x + 3)
\[ \int \frac {e^{-2+\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}} \left (18 x^8+6 x^9+e^{4 x} \left (-6+10 x+4 x^2\right )+e^2 \left (-36 x^3-9 x^4\right )+e^{3 x} \left (36 x^3+12 x^4\right )+e^{2 x} \left (36 x^4+48 x^5+12 x^6\right )+e^x \left (48 x^6+28 x^7+4 x^8\right )\right )}{243 x^3+162 x^4+27 x^5} \, dx=\int { \frac {{\left (6 \, x^{9} + 18 \, x^{8} - 9 \, {\left (x^{4} + 4 \, x^{3}\right )} e^{2} + 2 \, {\left (2 \, x^{2} + 5 \, x - 3\right )} e^{\left (4 \, x\right )} + 12 \, {\left (x^{4} + 3 \, x^{3}\right )} e^{\left (3 \, x\right )} + 12 \, {\left (x^{6} + 4 \, x^{5} + 3 \, x^{4}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{8} + 7 \, x^{7} + 12 \, x^{6}\right )} e^{x}\right )} e^{\left (\frac {{\left (x^{8} + 4 \, x^{6} e^{x} + 6 \, x^{4} e^{\left (2 \, x\right )} - 9 \, x^{3} e^{2} + 4 \, x^{2} e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )} e^{\left (-2\right )}}{9 \, x^{2}} - 2\right )}}{27 \, {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )}} \,d x } \]
integrate(((4*x^2+10*x-6)*exp(x)^4+(12*x^4+36*x^3)*exp(x)^3+(12*x^6+48*x^5 +36*x^4)*exp(x)^2+(4*x^8+28*x^7+48*x^6)*exp(x)+(-9*x^4-36*x^3)*exp(1)^2+6* x^9+18*x^8)*exp(1/9*(exp(x)^4+4*x^2*exp(x)^3+6*exp(x)^2*x^4+4*x^6*exp(x)-9 *x^3*exp(1)^2+x^8)/x^2/exp(1)^2)/(27*x^5+162*x^4+243*x^3)/exp(1)^2,x, algo rithm=\
integrate(1/27*(6*x^9 + 18*x^8 - 9*(x^4 + 4*x^3)*e^2 + 2*(2*x^2 + 5*x - 3) *e^(4*x) + 12*(x^4 + 3*x^3)*e^(3*x) + 12*(x^6 + 4*x^5 + 3*x^4)*e^(2*x) + 4 *(x^8 + 7*x^7 + 12*x^6)*e^x)*e^(1/9*(x^8 + 4*x^6*e^x + 6*x^4*e^(2*x) - 9*x ^3*e^2 + 4*x^2*e^(3*x) + e^(4*x))*e^(-2)/x^2 - 2)/(x^5 + 6*x^4 + 9*x^3), x )
Time = 13.54 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-2+\frac {e^{4 x}+4 e^{3 x} x^2-9 e^2 x^3+6 e^{2 x} x^4+4 e^x x^6+x^8}{9 e^2 x^2}} \left (18 x^8+6 x^9+e^{4 x} \left (-6+10 x+4 x^2\right )+e^2 \left (-36 x^3-9 x^4\right )+e^{3 x} \left (36 x^3+12 x^4\right )+e^{2 x} \left (36 x^4+48 x^5+12 x^6\right )+e^x \left (48 x^6+28 x^7+4 x^8\right )\right )}{243 x^3+162 x^4+27 x^5} \, dx=\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{-2}\,\left ({\mathrm {e}}^{4\,x}+4\,x^6\,{\mathrm {e}}^x+4\,x^2\,{\mathrm {e}}^{3\,x}+6\,x^4\,{\mathrm {e}}^{2\,x}-9\,x^3\,{\mathrm {e}}^2+x^8\right )}{9\,x^2}}}{3\,\left (x+3\right )} \]
int((exp(-2)*exp((exp(-2)*(exp(4*x)/9 + (4*x^6*exp(x))/9 + (4*x^2*exp(3*x) )/9 + (2*x^4*exp(2*x))/3 - x^3*exp(2) + x^8/9))/x^2)*(exp(4*x)*(10*x + 4*x ^2 - 6) + exp(x)*(48*x^6 + 28*x^7 + 4*x^8) + exp(3*x)*(36*x^3 + 12*x^4) - exp(2)*(36*x^3 + 9*x^4) + exp(2*x)*(36*x^4 + 48*x^5 + 12*x^6) + 18*x^8 + 6 *x^9))/(243*x^3 + 162*x^4 + 27*x^5),x)