Integrand size = 260, antiderivative size = 26 \[ \int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx=e^{e^{\left (-4+\frac {4 x^2}{4-x+\log (4)}\right )^2} x^2} \]
\[ \int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx=\int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx \]
Integrate[(E^(E^((256 - 128*x - 112*x^2 + 32*x^3 + 16*x^4 + (128 - 32*x - 32*x^2)*Log[4] + 16*Log[4]^2)/(16 - 8*x + x^2 + (8 - 2*x)*Log[4] + Log[4]^ 2))*x^2 + (256 - 128*x - 112*x^2 + 32*x^3 + 16*x^4 + (128 - 32*x - 32*x^2) *Log[4] + 16*Log[4]^2)/(16 - 8*x + x^2 + (8 - 2*x)*Log[4] + Log[4]^2))*(12 8*x - 96*x^2 - 1000*x^3 + 382*x^4 + 224*x^5 - 32*x^6 + (96*x - 48*x^2 - 50 6*x^3 + 96*x^4 + 64*x^5)*Log[4] + (24*x - 6*x^2 - 64*x^3)*Log[4]^2 + 2*x*L og[4]^3))/(64 - 48*x + 12*x^2 - x^3 + (48 - 24*x + 3*x^2)*Log[4] + (12 - 3 *x)*Log[4]^2 + Log[4]^3),x]
Integrate[(E^(E^((256 - 128*x - 112*x^2 + 32*x^3 + 16*x^4 + (128 - 32*x - 32*x^2)*Log[4] + 16*Log[4]^2)/(16 - 8*x + x^2 + (8 - 2*x)*Log[4] + Log[4]^ 2))*x^2 + (256 - 128*x - 112*x^2 + 32*x^3 + 16*x^4 + (128 - 32*x - 32*x^2) *Log[4] + 16*Log[4]^2)/(16 - 8*x + x^2 + (8 - 2*x)*Log[4] + Log[4]^2))*(12 8*x - 96*x^2 - 1000*x^3 + 382*x^4 + 224*x^5 - 32*x^6 + (96*x - 48*x^2 - 50 6*x^3 + 96*x^4 + 64*x^5)*Log[4] + (24*x - 6*x^2 - 64*x^3)*Log[4]^2 + 2*x*L og[4]^3))/(64 - 48*x + 12*x^2 - x^3 + (48 - 24*x + 3*x^2)*Log[4] + (12 - 3 *x)*Log[4]^2 + Log[4]^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.
| \(\displaystyle \int \frac {\left (-32 x^6+224 x^5+382 x^4-1000 x^3-96 x^2+\left (-64 x^3-6 x^2+24 x\right ) \log ^2(4)+\left (64 x^5+96 x^4-506 x^3-48 x^2+96 x\right ) \log (4)+128 x+2 x \log ^3(4)\right ) \exp \left (x^2 \exp \left (\frac {16 x^4+32 x^3-112 x^2+\left (-32 x^2-32 x+128\right ) \log (4)-128 x+256+16 \log ^2(4)}{x^2-8 x+(8-2 x) \log (4)+16+\log ^2(4)}\right )+\frac {16 x^4+32 x^3-112 x^2+\left (-32 x^2-32 x+128\right ) \log (4)-128 x+256+16 \log ^2(4)}{x^2-8 x+(8-2 x) \log (4)+16+\log ^2(4)}\right )}{-x^3+12 x^2+\left (3 x^2-24 x+48\right ) \log (4)-48 x+(12-3 x) \log ^2(4)+64+\log ^3(4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (-32 x^6+224 x^5+382 x^4-1000 x^3-96 x^2+\left (-64 x^3-6 x^2+24 x\right ) \log ^2(4)+\left (64 x^5+96 x^4-506 x^3-48 x^2+96 x\right ) \log (4)+x \left (128+2 \log ^3(4)\right )\right ) \exp \left (x^2 \exp \left (\frac {16 x^4+32 x^3-112 x^2+\left (-32 x^2-32 x+128\right ) \log (4)-128 x+256+16 \log ^2(4)}{x^2-8 x+(8-2 x) \log (4)+16+\log ^2(4)}\right )+\frac {16 x^4+32 x^3-112 x^2+\left (-32 x^2-32 x+128\right ) \log (4)-128 x+256+16 \log ^2(4)}{x^2-8 x+(8-2 x) \log (4)+16+\log ^2(4)}\right )}{-x^3+12 x^2+\left (3 x^2-24 x+48\right ) \log (4)-48 x+(12-3 x) \log ^2(4)+64+\log ^3(4)}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (-32 x^6+224 x^5+382 x^4-1000 x^3-96 x^2+\left (-64 x^3-6 x^2+24 x\right ) \log ^2(4)+\left (64 x^5+96 x^4-506 x^3-48 x^2+96 x\right ) \log (4)+x \left (128+2 \log ^3(4)\right )\right ) \exp \left (x^2 \exp \left (\frac {16 x^4+32 x^3-112 x^2+\left (-32 x^2-32 x+128\right ) \log (4)-128 x+256+16 \log ^2(4)}{x^2-8 x+(8-2 x) \log (4)+16+\log ^2(4)}\right )+\frac {16 x^4+32 x^3-112 x^2+\left (-32 x^2-32 x+128\right ) \log (4)-128 x+256+16 \log ^2(4)}{x^2-8 x+(8-2 x) \log (4)+16+\log ^2(4)}\right )}{(-x+4+\log (4))^3}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x \left (-16 x^5+16 x^4 (7+\log (16))+x^3 (191+48 \log (4))-x^2 \left (500+32 \log ^2(4)+253 \log (4)\right )-3 x (4+\log (4))^2+(4+\log (4))^3\right ) \exp \left (x^2 e^{\frac {16 \left (x^2+x-4-\log (4)\right )^2}{(-x+4+\log (4))^2}}+\frac {16 \left (x^2+x-4-\log (4)\right )^2}{(-x+4+\log (4))^2}\right )}{(-x+4+\log (4))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x \left (16 x^5-16 (7+\log (16)) x^4-(191+48 \log (4)) x^3+(4+\log (4)) (125+32 \log (4)) x^2+3 (4+\log (4))^2 x-(4+\log (4))^3\right )}{(-x+\log (4)+4)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x \left (16 x^5-16 (7+\log (16)) x^4-(191+48 \log (4)) x^3+(4+\log (4)) (125+32 \log (4)) x^2+3 (4+\log (4))^2 x-(4+\log (4))^3\right )}{(-x+\log (4)+4)^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-16 \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^3-16 \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) (5+\log (4)) x^2+\exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) \left (-1-96 \log ^2(4)-48 \log (4) (8-\log (16))+192 \log (16)\right ) x+\frac {16 \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) (4+\log (4))^3 (22+\log (1024))}{x-\log (4)-4}+\frac {16 \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) (4+\log (4))^4 (17+\log (256))}{(-x+\log (4)+4)^2}+16 \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) (4+\log (4))^2 (9+\log (16))+\frac {16 \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) (4+\log (4))^6}{(x-\log (4)-4)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (-\left (1+96 \log ^2(4)+48 \log (4) (8-\log (16))-192 \log (16)\right ) \int \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) xdx+16 (4+\log (4))^2 (9+\log (16)) \int \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right )dx-16 (5+\log (4)) \int \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2dx+16 (4+\log (4))^6 \int \frac {\exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right )}{(x-\log (4)-4)^3}dx+16 (4+\log (4))^3 (22+\log (1024)) \int \frac {\exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right )}{x-\log (4)-4}dx+16 (4+\log (4))^4 (17+\log (256)) \int \frac {\exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right )}{(-x+\log (4)+4)^2}dx-16 \int \exp \left (\exp \left (\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^2+\frac {16 \left (-x^2-x+\log (4)+4\right )^2}{(-x+\log (4)+4)^2}\right ) x^3dx\right )\) |
Int[(E^(E^((256 - 128*x - 112*x^2 + 32*x^3 + 16*x^4 + (128 - 32*x - 32*x^2 )*Log[4] + 16*Log[4]^2)/(16 - 8*x + x^2 + (8 - 2*x)*Log[4] + Log[4]^2))*x^ 2 + (256 - 128*x - 112*x^2 + 32*x^3 + 16*x^4 + (128 - 32*x - 32*x^2)*Log[4 ] + 16*Log[4]^2)/(16 - 8*x + x^2 + (8 - 2*x)*Log[4] + Log[4]^2))*(128*x - 96*x^2 - 1000*x^3 + 382*x^4 + 224*x^5 - 32*x^6 + (96*x - 48*x^2 - 506*x^3 + 96*x^4 + 64*x^5)*Log[4] + (24*x - 6*x^2 - 64*x^3)*Log[4]^2 + 2*x*Log[4]^ 3))/(64 - 48*x + 12*x^2 - x^3 + (48 - 24*x + 3*x^2)*Log[4] + (12 - 3*x)*Lo g[4]^2 + Log[4]^3),x]
3.23.77.3.1 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]]
Time = 10.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38
| method | result | size |
| risch | \({\mathrm e}^{x^{2} {\mathrm e}^{\frac {16 \left (-x^{2}+2 \ln \left (2\right )-x +4\right )^{2}}{\left (4+2 \ln \left (2\right )-x \right )^{2}}}}\) | \(36\) |
| parallelrisch | \({\mathrm e}^{x^{2} {\mathrm e}^{\frac {64 \ln \left (2\right )^{2}+2 \left (-32 x^{2}-32 x +128\right ) \ln \left (2\right )+16 x^{4}+32 x^{3}-112 x^{2}-128 x +256}{4 \ln \left (2\right )^{2}-4 x \ln \left (2\right )+x^{2}+16 \ln \left (2\right )-8 x +16}}}\) | \(73\) |
| norman | \(\frac {x^{2} {\mathrm e}^{x^{2} {\mathrm e}^{\frac {64 \ln \left (2\right )^{2}+2 \left (-32 x^{2}-32 x +128\right ) \ln \left (2\right )+16 x^{4}+32 x^{3}-112 x^{2}-128 x +256}{4 \ln \left (2\right )^{2}+2 \left (-2 x +8\right ) \ln \left (2\right )+x^{2}-8 x +16}}}+\left (4 \ln \left (2\right )^{2}+16 \ln \left (2\right )+16\right ) {\mathrm e}^{x^{2} {\mathrm e}^{\frac {64 \ln \left (2\right )^{2}+2 \left (-32 x^{2}-32 x +128\right ) \ln \left (2\right )+16 x^{4}+32 x^{3}-112 x^{2}-128 x +256}{4 \ln \left (2\right )^{2}+2 \left (-2 x +8\right ) \ln \left (2\right )+x^{2}-8 x +16}}}+\left (-4 \ln \left (2\right )-8\right ) x \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {64 \ln \left (2\right )^{2}+2 \left (-32 x^{2}-32 x +128\right ) \ln \left (2\right )+16 x^{4}+32 x^{3}-112 x^{2}-128 x +256}{4 \ln \left (2\right )^{2}+2 \left (-2 x +8\right ) \ln \left (2\right )+x^{2}-8 x +16}}}}{\left (4+2 \ln \left (2\right )-x \right )^{2}}\) | \(255\) |
int((16*x*ln(2)^3+4*(-64*x^3-6*x^2+24*x)*ln(2)^2+2*(64*x^5+96*x^4-506*x^3- 48*x^2+96*x)*ln(2)-32*x^6+224*x^5+382*x^4-1000*x^3-96*x^2+128*x)*exp((64*l n(2)^2+2*(-32*x^2-32*x+128)*ln(2)+16*x^4+32*x^3-112*x^2-128*x+256)/(4*ln(2 )^2+2*(-2*x+8)*ln(2)+x^2-8*x+16))*exp(x^2*exp((64*ln(2)^2+2*(-32*x^2-32*x+ 128)*ln(2)+16*x^4+32*x^3-112*x^2-128*x+256)/(4*ln(2)^2+2*(-2*x+8)*ln(2)+x^ 2-8*x+16)))/(8*ln(2)^3+4*(-3*x+12)*ln(2)^2+2*(3*x^2-24*x+48)*ln(2)-x^3+12* x^2-48*x+64),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 8.38 \[ \int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx=e^{\left (\frac {16 \, x^{4} + 32 \, x^{3} - 112 \, x^{2} + {\left (x^{4} + 4 \, x^{2} \log \left (2\right )^{2} - 8 \, x^{3} + 16 \, x^{2} - 4 \, {\left (x^{3} - 4 \, x^{2}\right )} \log \left (2\right )\right )} e^{\left (\frac {16 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 4 \, {\left (x^{2} + x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16\right )}}{x^{2} - 4 \, {\left (x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16}\right )} - 64 \, {\left (x^{2} + x - 4\right )} \log \left (2\right ) + 64 \, \log \left (2\right )^{2} - 128 \, x + 256}{x^{2} - 4 \, {\left (x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16} - \frac {16 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 4 \, {\left (x^{2} + x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16\right )}}{x^{2} - 4 \, {\left (x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16}\right )} \]
integrate((16*x*log(2)^3+4*(-64*x^3-6*x^2+24*x)*log(2)^2+2*(64*x^5+96*x^4- 506*x^3-48*x^2+96*x)*log(2)-32*x^6+224*x^5+382*x^4-1000*x^3-96*x^2+128*x)* exp((64*log(2)^2+2*(-32*x^2-32*x+128)*log(2)+16*x^4+32*x^3-112*x^2-128*x+2 56)/(4*log(2)^2+2*(-2*x+8)*log(2)+x^2-8*x+16))*exp(x^2*exp((64*log(2)^2+2* (-32*x^2-32*x+128)*log(2)+16*x^4+32*x^3-112*x^2-128*x+256)/(4*log(2)^2+2*( -2*x+8)*log(2)+x^2-8*x+16)))/(8*log(2)^3+4*(-3*x+12)*log(2)^2+2*(3*x^2-24* x+48)*log(2)-x^3+12*x^2-48*x+64),x, algorithm=\
e^((16*x^4 + 32*x^3 - 112*x^2 + (x^4 + 4*x^2*log(2)^2 - 8*x^3 + 16*x^2 - 4 *(x^3 - 4*x^2)*log(2))*e^(16*(x^4 + 2*x^3 - 7*x^2 - 4*(x^2 + x - 4)*log(2) + 4*log(2)^2 - 8*x + 16)/(x^2 - 4*(x - 4)*log(2) + 4*log(2)^2 - 8*x + 16) ) - 64*(x^2 + x - 4)*log(2) + 64*log(2)^2 - 128*x + 256)/(x^2 - 4*(x - 4)* log(2) + 4*log(2)^2 - 8*x + 16) - 16*(x^4 + 2*x^3 - 7*x^2 - 4*(x^2 + x - 4 )*log(2) + 4*log(2)^2 - 8*x + 16)/(x^2 - 4*(x - 4)*log(2) + 4*log(2)^2 - 8 *x + 16))
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (22) = 44\).
Time = 1.81 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx=e^{x^{2} e^{\frac {16 x^{4} + 32 x^{3} - 112 x^{2} - 128 x + \left (- 64 x^{2} - 64 x + 256\right ) \log {\left (2 \right )} + 64 \log {\left (2 \right )}^{2} + 256}{x^{2} - 8 x + \left (16 - 4 x\right ) \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2} + 16}}} \]
integrate((16*x*ln(2)**3+4*(-64*x**3-6*x**2+24*x)*ln(2)**2+2*(64*x**5+96*x **4-506*x**3-48*x**2+96*x)*ln(2)-32*x**6+224*x**5+382*x**4-1000*x**3-96*x* *2+128*x)*exp((64*ln(2)**2+2*(-32*x**2-32*x+128)*ln(2)+16*x**4+32*x**3-112 *x**2-128*x+256)/(4*ln(2)**2+2*(-2*x+8)*ln(2)+x**2-8*x+16))*exp(x**2*exp(( 64*ln(2)**2+2*(-32*x**2-32*x+128)*ln(2)+16*x**4+32*x**3-112*x**2-128*x+256 )/(4*ln(2)**2+2*(-2*x+8)*ln(2)+x**2-8*x+16)))/(8*ln(2)**3+4*(-3*x+12)*ln(2 )**2+2*(3*x**2-24*x+48)*ln(2)-x**3+12*x**2-48*x+64),x)
exp(x**2*exp((16*x**4 + 32*x**3 - 112*x**2 - 128*x + (-64*x**2 - 64*x + 25 6)*log(2) + 64*log(2)**2 + 256)/(x**2 - 8*x + (16 - 4*x)*log(2) + 4*log(2) **2 + 16)))
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (25) = 50\).
Time = 3.79 (sec) , antiderivative size = 226, normalized size of antiderivative = 8.69 \[ \int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx=e^{\left (28638903918474961204418783933674838490721739172170652529441449702311064005352904159345284265824628375429359509218999720074396860757073376700445026041564579620512874307979212102266801261478978776245040008231745247475930553606737583615358787106474295296 \, x^{2} e^{\left (\frac {256 \, \log \left (2\right )^{4}}{x^{2} - 4 \, x {\left (\log \left (2\right ) + 2\right )} + 4 \, \log \left (2\right )^{2} + 16 \, \log \left (2\right ) + 16} + 16 \, x^{2} + 64 \, x \log \left (2\right ) + 192 \, \log \left (2\right )^{2} + \frac {2048 \, \log \left (2\right )^{3}}{x^{2} - 4 \, x {\left (\log \left (2\right ) + 2\right )} + 4 \, \log \left (2\right )^{2} + 16 \, \log \left (2\right ) + 16} + \frac {512 \, \log \left (2\right )^{3}}{x - 2 \, \log \left (2\right ) - 4} + 160 \, x + \frac {6144 \, \log \left (2\right )^{2}}{x^{2} - 4 \, x {\left (\log \left (2\right ) + 2\right )} + 4 \, \log \left (2\right )^{2} + 16 \, \log \left (2\right ) + 16} + \frac {3200 \, \log \left (2\right )^{2}}{x - 2 \, \log \left (2\right ) - 4} + \frac {8192 \, \log \left (2\right )}{x^{2} - 4 \, x {\left (\log \left (2\right ) + 2\right )} + 4 \, \log \left (2\right )^{2} + 16 \, \log \left (2\right ) + 16} + \frac {6656 \, \log \left (2\right )}{x - 2 \, \log \left (2\right ) - 4} + \frac {4096}{x^{2} - 4 \, x {\left (\log \left (2\right ) + 2\right )} + 4 \, \log \left (2\right )^{2} + 16 \, \log \left (2\right ) + 16} + \frac {4608}{x - 2 \, \log \left (2\right ) - 4} + 912\right )}\right )} \]
integrate((16*x*log(2)^3+4*(-64*x^3-6*x^2+24*x)*log(2)^2+2*(64*x^5+96*x^4- 506*x^3-48*x^2+96*x)*log(2)-32*x^6+224*x^5+382*x^4-1000*x^3-96*x^2+128*x)* exp((64*log(2)^2+2*(-32*x^2-32*x+128)*log(2)+16*x^4+32*x^3-112*x^2-128*x+2 56)/(4*log(2)^2+2*(-2*x+8)*log(2)+x^2-8*x+16))*exp(x^2*exp((64*log(2)^2+2* (-32*x^2-32*x+128)*log(2)+16*x^4+32*x^3-112*x^2-128*x+256)/(4*log(2)^2+2*( -2*x+8)*log(2)+x^2-8*x+16)))/(8*log(2)^3+4*(-3*x+12)*log(2)^2+2*(3*x^2-24* x+48)*log(2)-x^3+12*x^2-48*x+64),x, algorithm=\
e^(28638903918474961204418783933674838490721739172170652529441449702311064 00535290415934528426582462837542935950921899972007439686075707337670044502 60415645796205128743079792121022668012614789787762450400082317452474759305 53606737583615358787106474295296*x^2*e^(256*log(2)^4/(x^2 - 4*x*(log(2) + 2) + 4*log(2)^2 + 16*log(2) + 16) + 16*x^2 + 64*x*log(2) + 192*log(2)^2 + 2048*log(2)^3/(x^2 - 4*x*(log(2) + 2) + 4*log(2)^2 + 16*log(2) + 16) + 512 *log(2)^3/(x - 2*log(2) - 4) + 160*x + 6144*log(2)^2/(x^2 - 4*x*(log(2) + 2) + 4*log(2)^2 + 16*log(2) + 16) + 3200*log(2)^2/(x - 2*log(2) - 4) + 819 2*log(2)/(x^2 - 4*x*(log(2) + 2) + 4*log(2)^2 + 16*log(2) + 16) + 6656*log (2)/(x - 2*log(2) - 4) + 4096/(x^2 - 4*x*(log(2) + 2) + 4*log(2)^2 + 16*lo g(2) + 16) + 4608/(x - 2*log(2) - 4) + 912))
\[ \int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx=\int { \frac {2 \, {\left (16 \, x^{6} - 112 \, x^{5} - 191 \, x^{4} - 8 \, x \log \left (2\right )^{3} + 500 \, x^{3} + 4 \, {\left (32 \, x^{3} + 3 \, x^{2} - 12 \, x\right )} \log \left (2\right )^{2} + 48 \, x^{2} - 2 \, {\left (32 \, x^{5} + 48 \, x^{4} - 253 \, x^{3} - 24 \, x^{2} + 48 \, x\right )} \log \left (2\right ) - 64 \, x\right )} e^{\left (x^{2} e^{\left (\frac {16 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 4 \, {\left (x^{2} + x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16\right )}}{x^{2} - 4 \, {\left (x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16}\right )} + \frac {16 \, {\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 4 \, {\left (x^{2} + x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16\right )}}{x^{2} - 4 \, {\left (x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 16}\right )}}{x^{3} + 12 \, {\left (x - 4\right )} \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} - 12 \, x^{2} - 6 \, {\left (x^{2} - 8 \, x + 16\right )} \log \left (2\right ) + 48 \, x - 64} \,d x } \]
integrate((16*x*log(2)^3+4*(-64*x^3-6*x^2+24*x)*log(2)^2+2*(64*x^5+96*x^4- 506*x^3-48*x^2+96*x)*log(2)-32*x^6+224*x^5+382*x^4-1000*x^3-96*x^2+128*x)* exp((64*log(2)^2+2*(-32*x^2-32*x+128)*log(2)+16*x^4+32*x^3-112*x^2-128*x+2 56)/(4*log(2)^2+2*(-2*x+8)*log(2)+x^2-8*x+16))*exp(x^2*exp((64*log(2)^2+2* (-32*x^2-32*x+128)*log(2)+16*x^4+32*x^3-112*x^2-128*x+256)/(4*log(2)^2+2*( -2*x+8)*log(2)+x^2-8*x+16)))/(8*log(2)^3+4*(-3*x+12)*log(2)^2+2*(3*x^2-24* x+48)*log(2)-x^3+12*x^2-48*x+64),x, algorithm=\
integrate(2*(16*x^6 - 112*x^5 - 191*x^4 - 8*x*log(2)^3 + 500*x^3 + 4*(32*x ^3 + 3*x^2 - 12*x)*log(2)^2 + 48*x^2 - 2*(32*x^5 + 48*x^4 - 253*x^3 - 24*x ^2 + 48*x)*log(2) - 64*x)*e^(x^2*e^(16*(x^4 + 2*x^3 - 7*x^2 - 4*(x^2 + x - 4)*log(2) + 4*log(2)^2 - 8*x + 16)/(x^2 - 4*(x - 4)*log(2) + 4*log(2)^2 - 8*x + 16)) + 16*(x^4 + 2*x^3 - 7*x^2 - 4*(x^2 + x - 4)*log(2) + 4*log(2)^ 2 - 8*x + 16)/(x^2 - 4*(x - 4)*log(2) + 4*log(2)^2 - 8*x + 16))/(x^3 + 12* (x - 4)*log(2)^2 - 8*log(2)^3 - 12*x^2 - 6*(x^2 - 8*x + 16)*log(2) + 48*x - 64), x)
Time = 28.11 (sec) , antiderivative size = 221, normalized size of antiderivative = 8.50 \[ \int \frac {e^{e^{\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} x^2+\frac {256-128 x-112 x^2+32 x^3+16 x^4+\left (128-32 x-32 x^2\right ) \log (4)+16 \log ^2(4)}{16-8 x+x^2+(8-2 x) \log (4)+\log ^2(4)}} \left (128 x-96 x^2-1000 x^3+382 x^4+224 x^5-32 x^6+\left (96 x-48 x^2-506 x^3+96 x^4+64 x^5\right ) \log (4)+\left (24 x-6 x^2-64 x^3\right ) \log ^2(4)+2 x \log ^3(4)\right )}{64-48 x+12 x^2-x^3+\left (48-24 x+3 x^2\right ) \log (4)+(12-3 x) \log ^2(4)+\log ^3(4)} \, dx={\mathrm {e}}^{{\left (\frac {1}{18446744073709551616}\right )}^{\frac {x^2+x-4}{16\,\ln \left (2\right )-8\,x-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+16}}\,x^2\,{\mathrm {e}}^{\frac {64\,{\ln \left (2\right )}^2}{16\,\ln \left (2\right )-8\,x-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+16}}\,{\mathrm {e}}^{\frac {16\,x^4}{16\,\ln \left (2\right )-8\,x-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+16}}\,{\mathrm {e}}^{\frac {32\,x^3}{16\,\ln \left (2\right )-8\,x-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+16}}\,{\mathrm {e}}^{-\frac {112\,x^2}{16\,\ln \left (2\right )-8\,x-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+16}}\,{\mathrm {e}}^{\frac {256}{16\,\ln \left (2\right )-8\,x-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+16}}\,{\mathrm {e}}^{-\frac {128\,x}{16\,\ln \left (2\right )-8\,x-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+16}}} \]
int((exp(x^2*exp((64*log(2)^2 - 2*log(2)*(32*x + 32*x^2 - 128) - 128*x - 1 12*x^2 + 32*x^3 + 16*x^4 + 256)/(4*log(2)^2 - 2*log(2)*(2*x - 8) - 8*x + x ^2 + 16)))*exp((64*log(2)^2 - 2*log(2)*(32*x + 32*x^2 - 128) - 128*x - 112 *x^2 + 32*x^3 + 16*x^4 + 256)/(4*log(2)^2 - 2*log(2)*(2*x - 8) - 8*x + x^2 + 16))*(128*x + 16*x*log(2)^3 - 4*log(2)^2*(6*x^2 - 24*x + 64*x^3) - 96*x ^2 - 1000*x^3 + 382*x^4 + 224*x^5 - 32*x^6 + 2*log(2)*(96*x - 48*x^2 - 506 *x^3 + 96*x^4 + 64*x^5)))/(2*log(2)*(3*x^2 - 24*x + 48) - 48*x - 4*log(2)^ 2*(3*x - 12) + 8*log(2)^3 + 12*x^2 - x^3 + 64),x)
exp((1/18446744073709551616)^((x + x^2 - 4)/(16*log(2) - 8*x - 4*x*log(2) + 4*log(2)^2 + x^2 + 16))*x^2*exp((64*log(2)^2)/(16*log(2) - 8*x - 4*x*log (2) + 4*log(2)^2 + x^2 + 16))*exp((16*x^4)/(16*log(2) - 8*x - 4*x*log(2) + 4*log(2)^2 + x^2 + 16))*exp((32*x^3)/(16*log(2) - 8*x - 4*x*log(2) + 4*lo g(2)^2 + x^2 + 16))*exp(-(112*x^2)/(16*log(2) - 8*x - 4*x*log(2) + 4*log(2 )^2 + x^2 + 16))*exp(256/(16*log(2) - 8*x - 4*x*log(2) + 4*log(2)^2 + x^2 + 16))*exp(-(128*x)/(16*log(2) - 8*x - 4*x*log(2) + 4*log(2)^2 + x^2 + 16) ))