Integrand size = 227, antiderivative size = 31 \[ \int \frac {2 x+e^{e^5} x+25 e^{-2 e^{1+x}} \left (-4 e^{1+3 e^5+x}+e^{1+2 e^5+x} (-24+12 x)+e^{1+e^5+x} \left (-48+48 x-12 x^2\right )+e^{1+x} \left (-32+48 x-24 x^2+4 x^3\right )\right )+5 e^{-e^{1+x}} \left (8-4 x+e^{2 e^5} \left (2-2 e^{1+x} x\right )+e^{1+x} \left (-8 x+8 x^2-2 x^3\right )+e^{e^5} \left (8-2 x+e^{1+x} \left (-8 x+4 x^2\right )\right )\right )}{16+2 e^{3 e^5}+e^{2 e^5} (12-6 x)-24 x+12 x^2-2 x^3+e^{e^5} \left (24-24 x+6 x^2\right )} \, dx=\left (5 e^{-e^{1+x}}-\frac {x}{2 \left (-2-e^{e^5}+x\right )}\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(31)=62\).
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.77 \[ \int \frac {2 x+e^{e^5} x+25 e^{-2 e^{1+x}} \left (-4 e^{1+3 e^5+x}+e^{1+2 e^5+x} (-24+12 x)+e^{1+e^5+x} \left (-48+48 x-12 x^2\right )+e^{1+x} \left (-32+48 x-24 x^2+4 x^3\right )\right )+5 e^{-e^{1+x}} \left (8-4 x+e^{2 e^5} \left (2-2 e^{1+x} x\right )+e^{1+x} \left (-8 x+8 x^2-2 x^3\right )+e^{e^5} \left (8-2 x+e^{1+x} \left (-8 x+4 x^2\right )\right )\right )}{16+2 e^{3 e^5}+e^{2 e^5} (12-6 x)-24 x+12 x^2-2 x^3+e^{e^5} \left (24-24 x+6 x^2\right )} \, dx=\frac {1}{2} \left (50 e^{-2 e^{1+x}}+\frac {\left (2+e^{e^5}\right )^2}{2 \left (2+e^{e^5}-x\right )^2}-\frac {2+e^{e^5}}{2+e^{e^5}-x}+\frac {10 e^{-e^{1+x}} x}{2+e^{e^5}-x}\right ) \]
Integrate[(2*x + E^E^5*x + (25*(-4*E^(1 + 3*E^5 + x) + E^(1 + 2*E^5 + x)*( -24 + 12*x) + E^(1 + E^5 + x)*(-48 + 48*x - 12*x^2) + E^(1 + x)*(-32 + 48* x - 24*x^2 + 4*x^3)))/E^(2*E^(1 + x)) + (5*(8 - 4*x + E^(2*E^5)*(2 - 2*E^( 1 + x)*x) + E^(1 + x)*(-8*x + 8*x^2 - 2*x^3) + E^E^5*(8 - 2*x + E^(1 + x)* (-8*x + 4*x^2))))/E^E^(1 + x))/(16 + 2*E^(3*E^5) + E^(2*E^5)*(12 - 6*x) - 24*x + 12*x^2 - 2*x^3 + E^E^5*(24 - 24*x + 6*x^2)),x]
(50/E^(2*E^(1 + x)) + (2 + E^E^5)^2/(2*(2 + E^E^5 - x)^2) - (2 + E^E^5)/(2 + E^E^5 - x) + (10*x)/(E^E^(1 + x)*(2 + E^E^5 - x)))/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.
| \(\displaystyle \int \frac {25 e^{-2 e^{x+1}} \left (e^{x+e^5+1} \left (-12 x^2+48 x-48\right )+e^{x+1} \left (4 x^3-24 x^2+48 x-32\right )+e^{x+2 e^5+1} (12 x-24)-4 e^{x+3 e^5+1}\right )+5 e^{-e^{x+1}} \left (e^{e^5} \left (e^{x+1} \left (4 x^2-8 x\right )-2 x+8\right )+e^{x+1} \left (-2 x^3+8 x^2-8 x\right )-4 x+e^{2 e^5} \left (2-2 e^{x+1} x\right )+8\right )+e^{e^5} x+2 x}{-2 x^3+12 x^2+e^{e^5} \left (6 x^2-24 x+24\right )-24 x+e^{2 e^5} (12-6 x)+2 e^{3 e^5}+16} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {25 e^{-2 e^{x+1}} \left (e^{x+e^5+1} \left (-12 x^2+48 x-48\right )+e^{x+1} \left (4 x^3-24 x^2+48 x-32\right )+e^{x+2 e^5+1} (12 x-24)-4 e^{x+3 e^5+1}\right )+5 e^{-e^{x+1}} \left (e^{e^5} \left (e^{x+1} \left (4 x^2-8 x\right )-2 x+8\right )+e^{x+1} \left (-2 x^3+8 x^2-8 x\right )-4 x+e^{2 e^5} \left (2-2 e^{x+1} x\right )+8\right )+\left (2+e^{e^5}\right ) x}{-2 x^3+12 x^2+e^{e^5} \left (6 x^2-24 x+24\right )-24 x+e^{2 e^5} (12-6 x)+2 e^{3 e^5}+16}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {25 e^{-2 e^{x+1}} \left (e^{x+e^5+1} \left (-12 x^2+48 x-48\right )+e^{x+1} \left (4 x^3-24 x^2+48 x-32\right )+e^{x+2 e^5+1} (12 x-24)-4 e^{x+3 e^5+1}\right )+5 e^{-e^{x+1}} \left (e^{e^5} \left (e^{x+1} \left (4 x^2-8 x\right )-2 x+8\right )+e^{x+1} \left (-2 x^3+8 x^2-8 x\right )-4 x+e^{2 e^5} \left (2-2 e^{x+1} x\right )+8\right )+\left (2+e^{e^5}\right ) x}{\left (\sqrt [3]{2} \left (2+e^{e^5}\right )-\sqrt [3]{2} x\right )^3}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-2 e^{x+1}} \left (10 e^{x+1} (x-2)^2-20 e^{x+e^5+1} (x-2)+10 e^{x+2 e^5+1}-2 \left (1+\frac {e^{e^5}}{2}\right ) e^{e^{x+1}}\right ) \left (-e^{e^{x+1}} x+10 x-20 \left (1+\frac {e^{e^5}}{2}\right )\right )}{2 \left (-x+e^{e^5}+2\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {e^{-2 e^{x+1}} \left (10 e^{x+1} (2-x)^2+20 e^{x+e^5+1} (2-x)+10 e^{x+2 e^5+1}-e^{e^{x+1}} \left (2+e^{e^5}\right )\right ) \left (e^{e^{x+1}} x-10 x+10 \left (2+e^{e^5}\right )\right )}{\left (-x+e^{e^5}+2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {e^{-2 e^{x+1}} \left (10 e^{x+1} (2-x)^2+20 e^{x+e^5+1} (2-x)+10 e^{x+2 e^5+1}-e^{e^{x+1}} \left (2+e^{e^5}\right )\right ) \left (e^{e^{x+1}} x-10 x+10 \left (2+e^{e^5}\right )\right )}{\left (-x+e^{e^5}+2\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {e^{-e^{x+1}} \left (2+e^{e^5}\right ) \left (-e^{e^{x+1}} x+10 x-20 \left (1+\frac {e^{e^5}}{2}\right )\right )}{\left (-x+e^{e^5}+2\right )^3}+\frac {10 e^{x-2 e^{x+1}+1} \left (e^{e^{x+1}} x-10 x+20 \left (1+\frac {e^{e^5}}{2}\right )\right )}{-x+e^{e^5}+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (10 \left (2+e^{e^5}\right ) \int \frac {e^{-e^{x+1}}}{\left (-x+e^{e^5}+2\right )^2}dx-10 \left (2+e^{e^5}\right ) \int \frac {e^{x-e^{x+1}+1}}{-x+e^{e^5}+2}dx+\frac {x^2}{2 \left (-x+e^{e^5}+2\right )^2}+50 e^{-2 e^{x+1}}-10 e^{-e^{x+1}}\right )\) |
Int[(2*x + E^E^5*x + (25*(-4*E^(1 + 3*E^5 + x) + E^(1 + 2*E^5 + x)*(-24 + 12*x) + E^(1 + E^5 + x)*(-48 + 48*x - 12*x^2) + E^(1 + x)*(-32 + 48*x - 24 *x^2 + 4*x^3)))/E^(2*E^(1 + x)) + (5*(8 - 4*x + E^(2*E^5)*(2 - 2*E^(1 + x) *x) + E^(1 + x)*(-8*x + 8*x^2 - 2*x^3) + E^E^5*(8 - 2*x + E^(1 + x)*(-8*x + 4*x^2))))/E^E^(1 + x))/(16 + 2*E^(3*E^5) + E^(2*E^5)*(12 - 6*x) - 24*x + 12*x^2 - 2*x^3 + E^E^5*(24 - 24*x + 6*x^2)),x]
3.18.17.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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[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. \(80\) vs. \(2(28)=56\).
Time = 0.89 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.61
| method | result | size |
| risch | \(\frac {\left (1+\frac {{\mathrm e}^{{\mathrm e}^{5}}}{2}\right ) x -\frac {{\mathrm e}^{2 \,{\mathrm e}^{5}}}{4}-{\mathrm e}^{{\mathrm e}^{5}}-1}{{\mathrm e}^{2 \,{\mathrm e}^{5}}-2 x \,{\mathrm e}^{{\mathrm e}^{5}}+x^{2}+4 \,{\mathrm e}^{{\mathrm e}^{5}}-4 x +4}+25 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}}+\frac {5 x \,{\mathrm e}^{-{\mathrm e}^{1+x}}}{{\mathrm e}^{{\mathrm e}^{5}}-x +2}\) | \(81\) |
| norman | \(\frac {25 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}} x^{2}+\left (1+\frac {{\mathrm e}^{{\mathrm e}^{5}}}{2}\right ) x +25 \left ({\mathrm e}^{2 \,{\mathrm e}^{5}}+4 \,{\mathrm e}^{{\mathrm e}^{5}}+4\right ) {\mathrm e}^{-2 \,{\mathrm e}^{1+x}}+25 \left (-2 \,{\mathrm e}^{{\mathrm e}^{5}}-4\right ) x \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}}+\left ({\mathrm e}^{{\mathrm e}^{5}}+2\right ) x \,{\mathrm e}^{-{\mathrm e}^{1+x}+\ln \left (5\right )}-x^{2} {\mathrm e}^{-{\mathrm e}^{1+x}+\ln \left (5\right )}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{5}}}{4}-{\mathrm e}^{{\mathrm e}^{5}}-1}{\left ({\mathrm e}^{{\mathrm e}^{5}}-x +2\right )^{2}}\) | \(129\) |
| parallelrisch | \(\frac {100 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}} {\mathrm e}^{2 \,{\mathrm e}^{5}}-200 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}} {\mathrm e}^{{\mathrm e}^{5}} x +100 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}} x^{2}-4+4 \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{-{\mathrm e}^{1+x}+\ln \left (5\right )} x +400 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}} {\mathrm e}^{{\mathrm e}^{5}}-4 x^{2} {\mathrm e}^{-{\mathrm e}^{1+x}+\ln \left (5\right )}-400 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}} x -{\mathrm e}^{2 \,{\mathrm e}^{5}}+2 x \,{\mathrm e}^{{\mathrm e}^{5}}+8 \,{\mathrm e}^{-{\mathrm e}^{1+x}+\ln \left (5\right )} x +400 \,{\mathrm e}^{-2 \,{\mathrm e}^{1+x}}-4 \,{\mathrm e}^{{\mathrm e}^{5}}+4 x}{4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}-8 x \,{\mathrm e}^{{\mathrm e}^{5}}+4 x^{2}+16 \,{\mathrm e}^{{\mathrm e}^{5}}-16 x +16}\) | \(196\) |
int(((-4*exp(1+x)*exp(exp(5))^3+(12*x-24)*exp(1+x)*exp(exp(5))^2+(-12*x^2+ 48*x-48)*exp(1+x)*exp(exp(5))+(4*x^3-24*x^2+48*x-32)*exp(1+x))*exp(-exp(1+ x)+ln(5))^2+((-2*x*exp(1+x)+2)*exp(exp(5))^2+((4*x^2-8*x)*exp(1+x)-2*x+8)* exp(exp(5))+(-2*x^3+8*x^2-8*x)*exp(1+x)-4*x+8)*exp(-exp(1+x)+ln(5))+x*exp( exp(5))+2*x)/(2*exp(exp(5))^3+(12-6*x)*exp(exp(5))^2+(6*x^2-24*x+24)*exp(e xp(5))-2*x^3+12*x^2-24*x+16),x,method=_RETURNVERBOSE)
((1+1/2*exp(exp(5)))*x-1/4*exp(2*exp(5))-exp(exp(5))-1)/(exp(2*exp(5))-2*x *exp(exp(5))+x^2+4*exp(exp(5))-4*x+4)+25*exp(-2*exp(1+x))+5*x/(exp(exp(5)) -x+2)*exp(-exp(1+x))
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.39 \[ \int \frac {2 x+e^{e^5} x+25 e^{-2 e^{1+x}} \left (-4 e^{1+3 e^5+x}+e^{1+2 e^5+x} (-24+12 x)+e^{1+e^5+x} \left (-48+48 x-12 x^2\right )+e^{1+x} \left (-32+48 x-24 x^2+4 x^3\right )\right )+5 e^{-e^{1+x}} \left (8-4 x+e^{2 e^5} \left (2-2 e^{1+x} x\right )+e^{1+x} \left (-8 x+8 x^2-2 x^3\right )+e^{e^5} \left (8-2 x+e^{1+x} \left (-8 x+4 x^2\right )\right )\right )}{16+2 e^{3 e^5}+e^{2 e^5} (12-6 x)-24 x+12 x^2-2 x^3+e^{e^5} \left (24-24 x+6 x^2\right )} \, dx=\frac {4 \, {\left (x^{2} - 2 \, {\left (x - 2\right )} e^{\left (e^{5}\right )} - 4 \, x + e^{\left (2 \, e^{5}\right )} + 4\right )} e^{\left (2 \, {\left (e^{\left (3 \, e^{5}\right )} \log \left (5\right ) - e^{\left (x + 3 \, e^{5} + 1\right )}\right )} e^{\left (-3 \, e^{5}\right )}\right )} - 4 \, {\left (x^{2} - x e^{\left (e^{5}\right )} - 2 \, x\right )} e^{\left ({\left (e^{\left (3 \, e^{5}\right )} \log \left (5\right ) - e^{\left (x + 3 \, e^{5} + 1\right )}\right )} e^{\left (-3 \, e^{5}\right )}\right )} + 2 \, {\left (x - 2\right )} e^{\left (e^{5}\right )} + 4 \, x - e^{\left (2 \, e^{5}\right )} - 4}{4 \, {\left (x^{2} - 2 \, {\left (x - 2\right )} e^{\left (e^{5}\right )} - 4 \, x + e^{\left (2 \, e^{5}\right )} + 4\right )}} \]
integrate(((-4*exp(1+x)*exp(exp(5))^3+(12*x-24)*exp(1+x)*exp(exp(5))^2+(-1 2*x^2+48*x-48)*exp(1+x)*exp(exp(5))+(4*x^3-24*x^2+48*x-32)*exp(1+x))*exp(- exp(1+x)+log(5))^2+((-2*x*exp(1+x)+2)*exp(exp(5))^2+((4*x^2-8*x)*exp(1+x)- 2*x+8)*exp(exp(5))+(-2*x^3+8*x^2-8*x)*exp(1+x)-4*x+8)*exp(-exp(1+x)+log(5) )+x*exp(exp(5))+2*x)/(2*exp(exp(5))^3+(12-6*x)*exp(exp(5))^2+(6*x^2-24*x+2 4)*exp(exp(5))-2*x^3+12*x^2-24*x+16),x, algorithm=\
1/4*(4*(x^2 - 2*(x - 2)*e^(e^5) - 4*x + e^(2*e^5) + 4)*e^(2*(e^(3*e^5)*log (5) - e^(x + 3*e^5 + 1))*e^(-3*e^5)) - 4*(x^2 - x*e^(e^5) - 2*x)*e^((e^(3* e^5)*log(5) - e^(x + 3*e^5 + 1))*e^(-3*e^5)) + 2*(x - 2)*e^(e^5) + 4*x - e ^(2*e^5) - 4)/(x^2 - 2*(x - 2)*e^(e^5) - 4*x + e^(2*e^5) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).
Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.90 \[ \int \frac {2 x+e^{e^5} x+25 e^{-2 e^{1+x}} \left (-4 e^{1+3 e^5+x}+e^{1+2 e^5+x} (-24+12 x)+e^{1+e^5+x} \left (-48+48 x-12 x^2\right )+e^{1+x} \left (-32+48 x-24 x^2+4 x^3\right )\right )+5 e^{-e^{1+x}} \left (8-4 x+e^{2 e^5} \left (2-2 e^{1+x} x\right )+e^{1+x} \left (-8 x+8 x^2-2 x^3\right )+e^{e^5} \left (8-2 x+e^{1+x} \left (-8 x+4 x^2\right )\right )\right )}{16+2 e^{3 e^5}+e^{2 e^5} (12-6 x)-24 x+12 x^2-2 x^3+e^{e^5} \left (24-24 x+6 x^2\right )} \, dx=\frac {- 5 x e^{- e^{x + 1}} + \left (25 x - 25 e^{e^{5}} - 50\right ) e^{- 2 e^{x + 1}}}{x - e^{e^{5}} - 2} + \frac {\left (- e^{e^{5}} - 2\right ) \left (- 2 x + 2 + e^{e^{5}}\right )}{4 x^{2} + x \left (- 8 e^{e^{5}} - 16\right ) + 16 + 16 e^{e^{5}} + 4 e^{2 e^{5}}} \]
integrate(((-4*exp(1+x)*exp(exp(5))**3+(12*x-24)*exp(1+x)*exp(exp(5))**2+( -12*x**2+48*x-48)*exp(1+x)*exp(exp(5))+(4*x**3-24*x**2+48*x-32)*exp(1+x))* exp(-exp(1+x)+ln(5))**2+((-2*x*exp(1+x)+2)*exp(exp(5))**2+((4*x**2-8*x)*ex p(1+x)-2*x+8)*exp(exp(5))+(-2*x**3+8*x**2-8*x)*exp(1+x)-4*x+8)*exp(-exp(1+ x)+ln(5))+x*exp(exp(5))+2*x)/(2*exp(exp(5))**3+(12-6*x)*exp(exp(5))**2+(6* x**2-24*x+24)*exp(exp(5))-2*x**3+12*x**2-24*x+16),x)
(-5*x*exp(-exp(x + 1)) + (25*x - 25*exp(exp(5)) - 50)*exp(-2*exp(x + 1)))/ (x - exp(exp(5)) - 2) + (-exp(exp(5)) - 2)*(-2*x + 2 + exp(exp(5)))/(4*x** 2 + x*(-8*exp(exp(5)) - 16) + 16 + 16*exp(exp(5)) + 4*exp(2*exp(5)))
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (29) = 58\).
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.68 \[ \int \frac {2 x+e^{e^5} x+25 e^{-2 e^{1+x}} \left (-4 e^{1+3 e^5+x}+e^{1+2 e^5+x} (-24+12 x)+e^{1+e^5+x} \left (-48+48 x-12 x^2\right )+e^{1+x} \left (-32+48 x-24 x^2+4 x^3\right )\right )+5 e^{-e^{1+x}} \left (8-4 x+e^{2 e^5} \left (2-2 e^{1+x} x\right )+e^{1+x} \left (-8 x+8 x^2-2 x^3\right )+e^{e^5} \left (8-2 x+e^{1+x} \left (-8 x+4 x^2\right )\right )\right )}{16+2 e^{3 e^5}+e^{2 e^5} (12-6 x)-24 x+12 x^2-2 x^3+e^{e^5} \left (24-24 x+6 x^2\right )} \, dx=-\frac {5 \, {\left (x e^{\left (e^{\left (x + 1\right )}\right )} - 5 \, x + 5 \, e^{\left (e^{5}\right )} + 10\right )} e^{\left (-2 \, e^{\left (x + 1\right )}\right )}}{x - e^{\left (e^{5}\right )} - 2} + \frac {{\left (2 \, x - e^{\left (e^{5}\right )} - 2\right )} e^{\left (e^{5}\right )}}{4 \, {\left (x^{2} - 2 \, x {\left (e^{\left (e^{5}\right )} + 2\right )} + e^{\left (2 \, e^{5}\right )} + 4 \, e^{\left (e^{5}\right )} + 4\right )}} + \frac {2 \, x - e^{\left (e^{5}\right )} - 2}{2 \, {\left (x^{2} - 2 \, x {\left (e^{\left (e^{5}\right )} + 2\right )} + e^{\left (2 \, e^{5}\right )} + 4 \, e^{\left (e^{5}\right )} + 4\right )}} \]
integrate(((-4*exp(1+x)*exp(exp(5))^3+(12*x-24)*exp(1+x)*exp(exp(5))^2+(-1 2*x^2+48*x-48)*exp(1+x)*exp(exp(5))+(4*x^3-24*x^2+48*x-32)*exp(1+x))*exp(- exp(1+x)+log(5))^2+((-2*x*exp(1+x)+2)*exp(exp(5))^2+((4*x^2-8*x)*exp(1+x)- 2*x+8)*exp(exp(5))+(-2*x^3+8*x^2-8*x)*exp(1+x)-4*x+8)*exp(-exp(1+x)+log(5) )+x*exp(exp(5))+2*x)/(2*exp(exp(5))^3+(12-6*x)*exp(exp(5))^2+(6*x^2-24*x+2 4)*exp(exp(5))-2*x^3+12*x^2-24*x+16),x, algorithm=\
-5*(x*e^(e^(x + 1)) - 5*x + 5*e^(e^5) + 10)*e^(-2*e^(x + 1))/(x - e^(e^5) - 2) + 1/4*(2*x - e^(e^5) - 2)*e^(e^5)/(x^2 - 2*x*(e^(e^5) + 2) + e^(2*e^5 ) + 4*e^(e^5) + 4) + 1/2*(2*x - e^(e^5) - 2)/(x^2 - 2*x*(e^(e^5) + 2) + e^ (2*e^5) + 4*e^(e^5) + 4)
\[ \int \frac {2 x+e^{e^5} x+25 e^{-2 e^{1+x}} \left (-4 e^{1+3 e^5+x}+e^{1+2 e^5+x} (-24+12 x)+e^{1+e^5+x} \left (-48+48 x-12 x^2\right )+e^{1+x} \left (-32+48 x-24 x^2+4 x^3\right )\right )+5 e^{-e^{1+x}} \left (8-4 x+e^{2 e^5} \left (2-2 e^{1+x} x\right )+e^{1+x} \left (-8 x+8 x^2-2 x^3\right )+e^{e^5} \left (8-2 x+e^{1+x} \left (-8 x+4 x^2\right )\right )\right )}{16+2 e^{3 e^5}+e^{2 e^5} (12-6 x)-24 x+12 x^2-2 x^3+e^{e^5} \left (24-24 x+6 x^2\right )} \, dx=\int { \frac {2 \, {\left ({\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{\left (x + 1\right )} + {\left (x e^{\left (x + 1\right )} - 1\right )} e^{\left (2 \, e^{5}\right )} - {\left (2 \, {\left (x^{2} - 2 \, x\right )} e^{\left (x + 1\right )} - x + 4\right )} e^{\left (e^{5}\right )} + 2 \, x - 4\right )} e^{\left (-e^{\left (x + 1\right )} + \log \left (5\right )\right )} - 4 \, {\left (3 \, {\left (x - 2\right )} e^{\left (x + 2 \, e^{5} + 1\right )} - 3 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x + e^{5} + 1\right )} + {\left (x^{3} - 6 \, x^{2} + 12 \, x - 8\right )} e^{\left (x + 1\right )} - e^{\left (x + 3 \, e^{5} + 1\right )}\right )} e^{\left (-2 \, e^{\left (x + 1\right )} + 2 \, \log \left (5\right )\right )} - x e^{\left (e^{5}\right )} - 2 \, x}{2 \, {\left (x^{3} - 6 \, x^{2} + 3 \, {\left (x - 2\right )} e^{\left (2 \, e^{5}\right )} - 3 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (e^{5}\right )} + 12 \, x - e^{\left (3 \, e^{5}\right )} - 8\right )}} \,d x } \]
integrate(((-4*exp(1+x)*exp(exp(5))^3+(12*x-24)*exp(1+x)*exp(exp(5))^2+(-1 2*x^2+48*x-48)*exp(1+x)*exp(exp(5))+(4*x^3-24*x^2+48*x-32)*exp(1+x))*exp(- exp(1+x)+log(5))^2+((-2*x*exp(1+x)+2)*exp(exp(5))^2+((4*x^2-8*x)*exp(1+x)- 2*x+8)*exp(exp(5))+(-2*x^3+8*x^2-8*x)*exp(1+x)-4*x+8)*exp(-exp(1+x)+log(5) )+x*exp(exp(5))+2*x)/(2*exp(exp(5))^3+(12-6*x)*exp(exp(5))^2+(6*x^2-24*x+2 4)*exp(exp(5))-2*x^3+12*x^2-24*x+16),x, algorithm=\
integrate(1/2*(2*((x^3 - 4*x^2 + 4*x)*e^(x + 1) + (x*e^(x + 1) - 1)*e^(2*e ^5) - (2*(x^2 - 2*x)*e^(x + 1) - x + 4)*e^(e^5) + 2*x - 4)*e^(-e^(x + 1) + log(5)) - 4*(3*(x - 2)*e^(x + 2*e^5 + 1) - 3*(x^2 - 4*x + 4)*e^(x + e^5 + 1) + (x^3 - 6*x^2 + 12*x - 8)*e^(x + 1) - e^(x + 3*e^5 + 1))*e^(-2*e^(x + 1) + 2*log(5)) - x*e^(e^5) - 2*x)/(x^3 - 6*x^2 + 3*(x - 2)*e^(2*e^5) - 3* (x^2 - 4*x + 4)*e^(e^5) + 12*x - e^(3*e^5) - 8), x)
Time = 11.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {2 x+e^{e^5} x+25 e^{-2 e^{1+x}} \left (-4 e^{1+3 e^5+x}+e^{1+2 e^5+x} (-24+12 x)+e^{1+e^5+x} \left (-48+48 x-12 x^2\right )+e^{1+x} \left (-32+48 x-24 x^2+4 x^3\right )\right )+5 e^{-e^{1+x}} \left (8-4 x+e^{2 e^5} \left (2-2 e^{1+x} x\right )+e^{1+x} \left (-8 x+8 x^2-2 x^3\right )+e^{e^5} \left (8-2 x+e^{1+x} \left (-8 x+4 x^2\right )\right )\right )}{16+2 e^{3 e^5}+e^{2 e^5} (12-6 x)-24 x+12 x^2-2 x^3+e^{e^5} \left (24-24 x+6 x^2\right )} \, dx=25\,{\mathrm {e}}^{-2\,\mathrm {e}\,{\mathrm {e}}^x}+\frac {x\,\left ({\mathrm {e}}^{{\mathrm {e}}^5}+2\right )-\frac {{\left ({\mathrm {e}}^{{\mathrm {e}}^5}+2\right )}^2}{2}}{2\,x^2+\left (-4\,{\mathrm {e}}^{{\mathrm {e}}^5}-8\right )\,x+2\,{\mathrm {e}}^{2\,{\mathrm {e}}^5}+8\,{\mathrm {e}}^{{\mathrm {e}}^5}+8}+\frac {5\,x\,{\mathrm {e}}^{-\mathrm {e}\,{\mathrm {e}}^x}}{{\mathrm {e}}^{{\mathrm {e}}^5}-x+2} \]
int((2*x - exp(log(5) - exp(x + 1))*(4*x + exp(x + 1)*(8*x - 8*x^2 + 2*x^3 ) + exp(exp(5))*(2*x + exp(x + 1)*(8*x - 4*x^2) - 8) + exp(2*exp(5))*(2*x* exp(x + 1) - 2) - 8) + x*exp(exp(5)) + exp(2*log(5) - 2*exp(x + 1))*(exp(x + 1)*(48*x - 24*x^2 + 4*x^3 - 32) - 4*exp(3*exp(5))*exp(x + 1) - exp(x + 1)*exp(exp(5))*(12*x^2 - 48*x + 48) + exp(2*exp(5))*exp(x + 1)*(12*x - 24) ))/(2*exp(3*exp(5)) - 24*x + exp(exp(5))*(6*x^2 - 24*x + 24) - exp(2*exp(5 ))*(6*x - 12) + 12*x^2 - 2*x^3 + 16),x)