Integrand size = 302, antiderivative size = 30 \[ \int \frac {e^{\frac {4}{2 x+x^2}} (-8-8 x)+8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x+8 x^3+2 x^4\right )+e^{e^4+2 e^{e^4+e^x-x}+e^x-x} \left (-8 x^2-8 x^3-2 x^4+e^x \left (8 x^2+8 x^3+2 x^4\right )\right )+e^{e^{e^4+e^x-x}} \left (e^{\frac {2}{2 x+x^2}} (-8-8 x)+8 x^2+8 x^3+2 x^4+e^{e^4+e^x-x} \left (-8 x^3-8 x^4-2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x^2-8 x^3-2 x^4\right )+e^x \left (8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (8 x^2+8 x^3+2 x^4\right )\right )\right )\right )}{4 x^2+4 x^3+x^4} \, dx=\left (e^{e^{e^4+e^x-x}}+e^{\frac {2}{x (2+x)}}+x\right )^2 \]
Time = 0.55 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\frac {4}{2 x+x^2}} (-8-8 x)+8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x+8 x^3+2 x^4\right )+e^{e^4+2 e^{e^4+e^x-x}+e^x-x} \left (-8 x^2-8 x^3-2 x^4+e^x \left (8 x^2+8 x^3+2 x^4\right )\right )+e^{e^{e^4+e^x-x}} \left (e^{\frac {2}{2 x+x^2}} (-8-8 x)+8 x^2+8 x^3+2 x^4+e^{e^4+e^x-x} \left (-8 x^3-8 x^4-2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x^2-8 x^3-2 x^4\right )+e^x \left (8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (8 x^2+8 x^3+2 x^4\right )\right )\right )\right )}{4 x^2+4 x^3+x^4} \, dx=e^{-\frac {2}{2+x}} \left (e^{\frac {1}{x}}+e^{e^{e^4+e^x-x}+\frac {1}{2+x}}+e^{\frac {1}{2+x}} x\right )^2 \]
Integrate[(E^(4/(2*x + x^2))*(-8 - 8*x) + 8*x^3 + 8*x^4 + 2*x^5 + E^(2/(2* x + x^2))*(-8*x + 8*x^3 + 2*x^4) + E^(E^4 + 2*E^(E^4 + E^x - x) + E^x - x) *(-8*x^2 - 8*x^3 - 2*x^4 + E^x*(8*x^2 + 8*x^3 + 2*x^4)) + E^E^(E^4 + E^x - x)*(E^(2/(2*x + x^2))*(-8 - 8*x) + 8*x^2 + 8*x^3 + 2*x^4 + E^(E^4 + E^x - x)*(-8*x^3 - 8*x^4 - 2*x^5 + E^(2/(2*x + x^2))*(-8*x^2 - 8*x^3 - 2*x^4) + E^x*(8*x^3 + 8*x^4 + 2*x^5 + E^(2/(2*x + x^2))*(8*x^2 + 8*x^3 + 2*x^4)))) )/(4*x^2 + 4*x^3 + x^4),x]
Time = 13.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2026, 2007, 7239, 27, 25, 7237}
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.
| \(\displaystyle \int \frac {2 x^5+8 x^4+8 x^3+e^{\frac {4}{x^2+2 x}} (-8 x-8)+e^{\frac {2}{x^2+2 x}} \left (2 x^4+8 x^3-8 x\right )+e^{-x+2 e^{-x+e^x+e^4}+e^x+e^4} \left (-2 x^4-8 x^3-8 x^2+e^x \left (2 x^4+8 x^3+8 x^2\right )\right )+e^{e^{-x+e^x+e^4}} \left (2 x^4+8 x^3+8 x^2+e^{\frac {2}{x^2+2 x}} (-8 x-8)+e^{-x+e^x+e^4} \left (-2 x^5-8 x^4-8 x^3+e^{\frac {2}{x^2+2 x}} \left (-2 x^4-8 x^3-8 x^2\right )+e^x \left (2 x^5+8 x^4+8 x^3+e^{\frac {2}{x^2+2 x}} \left (2 x^4+8 x^3+8 x^2\right )\right )\right )\right )}{x^4+4 x^3+4 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^5+8 x^4+8 x^3+e^{\frac {4}{x^2+2 x}} (-8 x-8)+e^{\frac {2}{x^2+2 x}} \left (2 x^4+8 x^3-8 x\right )+e^{-x+2 e^{-x+e^x+e^4}+e^x+e^4} \left (-2 x^4-8 x^3-8 x^2+e^x \left (2 x^4+8 x^3+8 x^2\right )\right )+e^{e^{-x+e^x+e^4}} \left (2 x^4+8 x^3+8 x^2+e^{\frac {2}{x^2+2 x}} (-8 x-8)+e^{-x+e^x+e^4} \left (-2 x^5-8 x^4-8 x^3+e^{\frac {2}{x^2+2 x}} \left (-2 x^4-8 x^3-8 x^2\right )+e^x \left (2 x^5+8 x^4+8 x^3+e^{\frac {2}{x^2+2 x}} \left (2 x^4+8 x^3+8 x^2\right )\right )\right )\right )}{x^2 \left (x^2+4 x+4\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {2 x^5+8 x^4+8 x^3+e^{\frac {4}{x^2+2 x}} (-8 x-8)+e^{\frac {2}{x^2+2 x}} \left (2 x^4+8 x^3-8 x\right )+e^{-x+2 e^{-x+e^x+e^4}+e^x+e^4} \left (-2 x^4-8 x^3-8 x^2+e^x \left (2 x^4+8 x^3+8 x^2\right )\right )+e^{e^{-x+e^x+e^4}} \left (2 x^4+8 x^3+8 x^2+e^{\frac {2}{x^2+2 x}} (-8 x-8)+e^{-x+e^x+e^4} \left (-2 x^5-8 x^4-8 x^3+e^{\frac {2}{x^2+2 x}} \left (-2 x^4-8 x^3-8 x^2\right )+e^x \left (2 x^5+8 x^4+8 x^3+e^{\frac {2}{x^2+2 x}} \left (2 x^4+8 x^3+8 x^2\right )\right )\right )\right )}{x^2 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{-x} \left (e^{\frac {2}{x^2+2 x}}+x+e^{e^{-x+e^x+e^4}}\right ) \left (-e^{e^{-x+e^x+e^4}+e^x+e^4} x^2 (x+2)^2+e^x x^2 (x+2)^2+e^{x+e^{-x+e^x+e^4}+e^x+e^4} x^2 (x+2)^2-4 e^{\frac {2}{x^2+2 x}+x} (x+1)\right )}{x^2 (x+2)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {e^{-x} \left (x+e^{e^{-x+e^x+e^4}}+e^{\frac {2}{x^2+2 x}}\right ) \left (e^{e^4+e^{-x+e^x+e^4}+e^x} x^2 (x+2)^2-e^x x^2 (x+2)^2-e^{x+e^{-x+e^x+e^4}+e^x+e^4} x^2 (x+2)^2+4 e^{x+\frac {2}{x^2+2 x}} (x+1)\right )}{x^2 (x+2)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {e^{-x} \left (x+e^{e^{-x+e^x+e^4}}+e^{\frac {2}{x^2+2 x}}\right ) \left (e^{e^4+e^{-x+e^x+e^4}+e^x} x^2 (x+2)^2-e^x x^2 (x+2)^2-e^{x+e^{-x+e^x+e^4}+e^x+e^4} x^2 (x+2)^2+4 e^{x+\frac {2}{x^2+2 x}} (x+1)\right )}{x^2 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \left (e^{\frac {2}{x^2+2 x}}+x+e^{e^{-x+e^x+e^4}}\right )^2\) |
Int[(E^(4/(2*x + x^2))*(-8 - 8*x) + 8*x^3 + 8*x^4 + 2*x^5 + E^(2/(2*x + x^ 2))*(-8*x + 8*x^3 + 2*x^4) + E^(E^4 + 2*E^(E^4 + E^x - x) + E^x - x)*(-8*x ^2 - 8*x^3 - 2*x^4 + E^x*(8*x^2 + 8*x^3 + 2*x^4)) + E^E^(E^4 + E^x - x)*(E ^(2/(2*x + x^2))*(-8 - 8*x) + 8*x^2 + 8*x^3 + 2*x^4 + E^(E^4 + E^x - x)*(- 8*x^3 - 8*x^4 - 2*x^5 + E^(2/(2*x + x^2))*(-8*x^2 - 8*x^3 - 2*x^4) + E^x*( 8*x^3 + 8*x^4 + 2*x^5 + E^(2/(2*x + x^2))*(8*x^2 + 8*x^3 + 2*x^4)))))/(4*x ^2 + 4*x^3 + x^4),x]
3.8.92.3.1 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_.)*(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])
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
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. \(69\) vs. \(2(25)=50\).
Time = 217.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33
| method | result | size |
| risch | \(x^{2}+2 x \,{\mathrm e}^{\frac {2}{x \left (2+x \right )}}+{\mathrm e}^{\frac {4}{x \left (2+x \right )}}+{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{4}-x}}+\left (2 x +2 \,{\mathrm e}^{\frac {2}{x \left (2+x \right )}}\right ) {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{4}-x}}\) | \(70\) |
| parallelrisch | \(x^{2}+2 x \,{\mathrm e}^{\frac {2}{x \left (2+x \right )}}+2 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{4}-x}} x +{\mathrm e}^{\frac {4}{x \left (2+x \right )}}+2 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{4}-x}} {\mathrm e}^{\frac {2}{x \left (2+x \right )}}+{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{4}-x}}-4\) | \(81\) |
int((((2*x^4+8*x^3+8*x^2)*exp(x)-2*x^4-8*x^3-8*x^2)*exp(exp(x)+exp(4)-x)*e xp(exp(exp(x)+exp(4)-x))^2+((((2*x^4+8*x^3+8*x^2)*exp(2/(x^2+2*x))+2*x^5+8 *x^4+8*x^3)*exp(x)+(-2*x^4-8*x^3-8*x^2)*exp(2/(x^2+2*x))-2*x^5-8*x^4-8*x^3 )*exp(exp(x)+exp(4)-x)+(-8*x-8)*exp(2/(x^2+2*x))+2*x^4+8*x^3+8*x^2)*exp(ex p(exp(x)+exp(4)-x))+(-8*x-8)*exp(2/(x^2+2*x))^2+(2*x^4+8*x^3-8*x)*exp(2/(x ^2+2*x))+2*x^5+8*x^4+8*x^3)/(x^4+4*x^3+4*x^2),x,method=_RETURNVERBOSE)
x^2+2*x*exp(2/x/(2+x))+exp(4/x/(2+x))+exp(2*exp(exp(x)+exp(4)-x))+(2*x+2*e xp(2/x/(2+x)))*exp(exp(exp(x)+exp(4)-x))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30 \[ \int \frac {e^{\frac {4}{2 x+x^2}} (-8-8 x)+8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x+8 x^3+2 x^4\right )+e^{e^4+2 e^{e^4+e^x-x}+e^x-x} \left (-8 x^2-8 x^3-2 x^4+e^x \left (8 x^2+8 x^3+2 x^4\right )\right )+e^{e^{e^4+e^x-x}} \left (e^{\frac {2}{2 x+x^2}} (-8-8 x)+8 x^2+8 x^3+2 x^4+e^{e^4+e^x-x} \left (-8 x^3-8 x^4-2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x^2-8 x^3-2 x^4\right )+e^x \left (8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (8 x^2+8 x^3+2 x^4\right )\right )\right )\right )}{4 x^2+4 x^3+x^4} \, dx=x^{2} + 2 \, x e^{\left (\frac {2}{x^{2} + 2 \, x}\right )} + 2 \, {\left (x + e^{\left (\frac {2}{x^{2} + 2 \, x}\right )}\right )} e^{\left (e^{\left (-x + e^{4} + e^{x}\right )}\right )} + e^{\left (2 \, e^{\left (-x + e^{4} + e^{x}\right )}\right )} + e^{\left (\frac {4}{x^{2} + 2 \, x}\right )} \]
integrate((((2*x^4+8*x^3+8*x^2)*exp(x)-2*x^4-8*x^3-8*x^2)*exp(exp(x)+exp(4 )-x)*exp(exp(exp(x)+exp(4)-x))^2+((((2*x^4+8*x^3+8*x^2)*exp(2/(x^2+2*x))+2 *x^5+8*x^4+8*x^3)*exp(x)+(-2*x^4-8*x^3-8*x^2)*exp(2/(x^2+2*x))-2*x^5-8*x^4 -8*x^3)*exp(exp(x)+exp(4)-x)+(-8*x-8)*exp(2/(x^2+2*x))+2*x^4+8*x^3+8*x^2)* exp(exp(exp(x)+exp(4)-x))+(-8*x-8)*exp(2/(x^2+2*x))^2+(2*x^4+8*x^3-8*x)*ex p(2/(x^2+2*x))+2*x^5+8*x^4+8*x^3)/(x^4+4*x^3+4*x^2),x, algorithm=\
x^2 + 2*x*e^(2/(x^2 + 2*x)) + 2*(x + e^(2/(x^2 + 2*x)))*e^(e^(-x + e^4 + e ^x)) + e^(2*e^(-x + e^4 + e^x)) + e^(4/(x^2 + 2*x))
Timed out. \[ \int \frac {e^{\frac {4}{2 x+x^2}} (-8-8 x)+8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x+8 x^3+2 x^4\right )+e^{e^4+2 e^{e^4+e^x-x}+e^x-x} \left (-8 x^2-8 x^3-2 x^4+e^x \left (8 x^2+8 x^3+2 x^4\right )\right )+e^{e^{e^4+e^x-x}} \left (e^{\frac {2}{2 x+x^2}} (-8-8 x)+8 x^2+8 x^3+2 x^4+e^{e^4+e^x-x} \left (-8 x^3-8 x^4-2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x^2-8 x^3-2 x^4\right )+e^x \left (8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (8 x^2+8 x^3+2 x^4\right )\right )\right )\right )}{4 x^2+4 x^3+x^4} \, dx=\text {Timed out} \]
integrate((((2*x**4+8*x**3+8*x**2)*exp(x)-2*x**4-8*x**3-8*x**2)*exp(exp(x) +exp(4)-x)*exp(exp(exp(x)+exp(4)-x))**2+((((2*x**4+8*x**3+8*x**2)*exp(2/(x **2+2*x))+2*x**5+8*x**4+8*x**3)*exp(x)+(-2*x**4-8*x**3-8*x**2)*exp(2/(x**2 +2*x))-2*x**5-8*x**4-8*x**3)*exp(exp(x)+exp(4)-x)+(-8*x-8)*exp(2/(x**2+2*x ))+2*x**4+8*x**3+8*x**2)*exp(exp(exp(x)+exp(4)-x))+(-8*x-8)*exp(2/(x**2+2* x))**2+(2*x**4+8*x**3-8*x)*exp(2/(x**2+2*x))+2*x**5+8*x**4+8*x**3)/(x**4+4 *x**3+4*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.13 \[ \int \frac {e^{\frac {4}{2 x+x^2}} (-8-8 x)+8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x+8 x^3+2 x^4\right )+e^{e^4+2 e^{e^4+e^x-x}+e^x-x} \left (-8 x^2-8 x^3-2 x^4+e^x \left (8 x^2+8 x^3+2 x^4\right )\right )+e^{e^{e^4+e^x-x}} \left (e^{\frac {2}{2 x+x^2}} (-8-8 x)+8 x^2+8 x^3+2 x^4+e^{e^4+e^x-x} \left (-8 x^3-8 x^4-2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x^2-8 x^3-2 x^4\right )+e^x \left (8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (8 x^2+8 x^3+2 x^4\right )\right )\right )\right )}{4 x^2+4 x^3+x^4} \, dx=x^{2} + {\left (2 \, x e^{\frac {1}{x}} + 2 \, {\left (x e^{\left (\frac {1}{x + 2}\right )} + e^{\frac {1}{x}}\right )} e^{\left (e^{\left (-x + e^{4} + e^{x}\right )}\right )} + e^{\left (\frac {1}{x + 2} + 2 \, e^{\left (-x + e^{4} + e^{x}\right )}\right )}\right )} e^{\left (-\frac {1}{x + 2}\right )} \]
integrate((((2*x^4+8*x^3+8*x^2)*exp(x)-2*x^4-8*x^3-8*x^2)*exp(exp(x)+exp(4 )-x)*exp(exp(exp(x)+exp(4)-x))^2+((((2*x^4+8*x^3+8*x^2)*exp(2/(x^2+2*x))+2 *x^5+8*x^4+8*x^3)*exp(x)+(-2*x^4-8*x^3-8*x^2)*exp(2/(x^2+2*x))-2*x^5-8*x^4 -8*x^3)*exp(exp(x)+exp(4)-x)+(-8*x-8)*exp(2/(x^2+2*x))+2*x^4+8*x^3+8*x^2)* exp(exp(exp(x)+exp(4)-x))+(-8*x-8)*exp(2/(x^2+2*x))^2+(2*x^4+8*x^3-8*x)*ex p(2/(x^2+2*x))+2*x^5+8*x^4+8*x^3)/(x^4+4*x^3+4*x^2),x, algorithm=\
x^2 + (2*x*e^(1/x) + 2*(x*e^(1/(x + 2)) + e^(1/x))*e^(e^(-x + e^4 + e^x)) + e^(1/(x + 2) + 2*e^(-x + e^4 + e^x)))*e^(-1/(x + 2))
\[ \int \frac {e^{\frac {4}{2 x+x^2}} (-8-8 x)+8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x+8 x^3+2 x^4\right )+e^{e^4+2 e^{e^4+e^x-x}+e^x-x} \left (-8 x^2-8 x^3-2 x^4+e^x \left (8 x^2+8 x^3+2 x^4\right )\right )+e^{e^{e^4+e^x-x}} \left (e^{\frac {2}{2 x+x^2}} (-8-8 x)+8 x^2+8 x^3+2 x^4+e^{e^4+e^x-x} \left (-8 x^3-8 x^4-2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x^2-8 x^3-2 x^4\right )+e^x \left (8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (8 x^2+8 x^3+2 x^4\right )\right )\right )\right )}{4 x^2+4 x^3+x^4} \, dx=\int { \frac {2 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3} - {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} e^{x}\right )} e^{\left (-x + e^{4} + e^{x} + 2 \, e^{\left (-x + e^{4} + e^{x}\right )}\right )} - 4 \, {\left (x + 1\right )} e^{\left (\frac {4}{x^{2} + 2 \, x}\right )} + {\left (x^{4} + 4 \, x^{3} - 4 \, x\right )} e^{\left (\frac {2}{x^{2} + 2 \, x}\right )} + {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2} - {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3} - {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3} + {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (\frac {2}{x^{2} + 2 \, x}\right )}\right )} e^{x} + {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (\frac {2}{x^{2} + 2 \, x}\right )}\right )} e^{\left (-x + e^{4} + e^{x}\right )} - 4 \, {\left (x + 1\right )} e^{\left (\frac {2}{x^{2} + 2 \, x}\right )}\right )} e^{\left (e^{\left (-x + e^{4} + e^{x}\right )}\right )}\right )}}{x^{4} + 4 \, x^{3} + 4 \, x^{2}} \,d x } \]
integrate((((2*x^4+8*x^3+8*x^2)*exp(x)-2*x^4-8*x^3-8*x^2)*exp(exp(x)+exp(4 )-x)*exp(exp(exp(x)+exp(4)-x))^2+((((2*x^4+8*x^3+8*x^2)*exp(2/(x^2+2*x))+2 *x^5+8*x^4+8*x^3)*exp(x)+(-2*x^4-8*x^3-8*x^2)*exp(2/(x^2+2*x))-2*x^5-8*x^4 -8*x^3)*exp(exp(x)+exp(4)-x)+(-8*x-8)*exp(2/(x^2+2*x))+2*x^4+8*x^3+8*x^2)* exp(exp(exp(x)+exp(4)-x))+(-8*x-8)*exp(2/(x^2+2*x))^2+(2*x^4+8*x^3-8*x)*ex p(2/(x^2+2*x))+2*x^5+8*x^4+8*x^3)/(x^4+4*x^3+4*x^2),x, algorithm=\
integrate(2*(x^5 + 4*x^4 + 4*x^3 - (x^4 + 4*x^3 + 4*x^2 - (x^4 + 4*x^3 + 4 *x^2)*e^x)*e^(-x + e^4 + e^x + 2*e^(-x + e^4 + e^x)) - 4*(x + 1)*e^(4/(x^2 + 2*x)) + (x^4 + 4*x^3 - 4*x)*e^(2/(x^2 + 2*x)) + (x^4 + 4*x^3 + 4*x^2 - (x^5 + 4*x^4 + 4*x^3 - (x^5 + 4*x^4 + 4*x^3 + (x^4 + 4*x^3 + 4*x^2)*e^(2/( x^2 + 2*x)))*e^x + (x^4 + 4*x^3 + 4*x^2)*e^(2/(x^2 + 2*x)))*e^(-x + e^4 + e^x) - 4*(x + 1)*e^(2/(x^2 + 2*x)))*e^(e^(-x + e^4 + e^x)))/(x^4 + 4*x^3 + 4*x^2), x)
Time = 8.68 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.50 \[ \int \frac {e^{\frac {4}{2 x+x^2}} (-8-8 x)+8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x+8 x^3+2 x^4\right )+e^{e^4+2 e^{e^4+e^x-x}+e^x-x} \left (-8 x^2-8 x^3-2 x^4+e^x \left (8 x^2+8 x^3+2 x^4\right )\right )+e^{e^{e^4+e^x-x}} \left (e^{\frac {2}{2 x+x^2}} (-8-8 x)+8 x^2+8 x^3+2 x^4+e^{e^4+e^x-x} \left (-8 x^3-8 x^4-2 x^5+e^{\frac {2}{2 x+x^2}} \left (-8 x^2-8 x^3-2 x^4\right )+e^x \left (8 x^3+8 x^4+2 x^5+e^{\frac {2}{2 x+x^2}} \left (8 x^2+8 x^3+2 x^4\right )\right )\right )\right )}{4 x^2+4 x^3+x^4} \, dx={\mathrm {e}}^{2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^4}}+{\mathrm {e}}^{\frac {4}{x^2+2\,x}}+2\,x\,{\mathrm {e}}^{\frac {2}{x^2+2\,x}}+{\mathrm {e}}^{{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,\left (2\,x+2\,{\mathrm {e}}^{\frac {2}{x^2+2\,x}}\right )+x^2 \]
int((exp(exp(exp(4) - x + exp(x)))*(8*x^2 - exp(2/(2*x + x^2))*(8*x + 8) - exp(exp(4) - x + exp(x))*(exp(2/(2*x + x^2))*(8*x^2 + 8*x^3 + 2*x^4) - ex p(x)*(exp(2/(2*x + x^2))*(8*x^2 + 8*x^3 + 2*x^4) + 8*x^3 + 8*x^4 + 2*x^5) + 8*x^3 + 8*x^4 + 2*x^5) + 8*x^3 + 2*x^4) + exp(2/(2*x + x^2))*(8*x^3 - 8* x + 2*x^4) - exp(4/(2*x + x^2))*(8*x + 8) + 8*x^3 + 8*x^4 + 2*x^5 - exp(2* exp(exp(4) - x + exp(x)))*exp(exp(4) - x + exp(x))*(8*x^2 - exp(x)*(8*x^2 + 8*x^3 + 2*x^4) + 8*x^3 + 2*x^4))/(4*x^2 + 4*x^3 + x^4),x)