Integrand size = 289, antiderivative size = 28 \[ \int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx=2 e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} (4+x) \]
\[ \int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx=\int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx \]
Integrate[(E^(-x - 4*Log[3] + Log[x + E^x*Log[x^2]])^(-1)*(E^x*(-16 - 4*x) - 8*x + 6*x^2 + 2*x^3 + 2*x^4 + 16*x^3*Log[3] + 32*x^2*Log[3]^2 + E^x*(2* x^3 + 16*x^2*Log[3] + 32*x*Log[3]^2)*Log[x^2] + (-4*x^3 - 16*x^2*Log[3] + E^x*(-4*x^2 - 16*x*Log[3])*Log[x^2])*Log[x + E^x*Log[x^2]] + (2*x^2 + 2*E^ x*x*Log[x^2])*Log[x + E^x*Log[x^2]]^2))/(x^4 + 8*x^3*Log[3] + 16*x^2*Log[3 ]^2 + E^x*(x^3 + 8*x^2*Log[3] + 16*x*Log[3]^2)*Log[x^2] + (-2*x^3 - 8*x^2* Log[3] + E^x*(-2*x^2 - 8*x*Log[3])*Log[x^2])*Log[x + E^x*Log[x^2]] + (x^2 + E^x*x*Log[x^2])*Log[x + E^x*Log[x^2]]^2),x]
Integrate[(E^(-x - 4*Log[3] + Log[x + E^x*Log[x^2]])^(-1)*(E^x*(-16 - 4*x) - 8*x + 6*x^2 + 2*x^3 + 2*x^4 + 16*x^3*Log[3] + 32*x^2*Log[3]^2 + E^x*(2* x^3 + 16*x^2*Log[3] + 32*x*Log[3]^2)*Log[x^2] + (-4*x^3 - 16*x^2*Log[3] + E^x*(-4*x^2 - 16*x*Log[3])*Log[x^2])*Log[x + E^x*Log[x^2]] + (2*x^2 + 2*E^ x*x*Log[x^2])*Log[x + E^x*Log[x^2]]^2))/(x^4 + 8*x^3*Log[3] + 16*x^2*Log[3 ]^2 + E^x*(x^3 + 8*x^2*Log[3] + 16*x*Log[3]^2)*Log[x^2] + (-2*x^3 - 8*x^2* Log[3] + E^x*(-2*x^2 - 8*x*Log[3])*Log[x^2])*Log[x + E^x*Log[x^2]] + (x^2 + E^x*x*Log[x^2])*Log[x + E^x*Log[x^2]]^2), 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.
\(\displaystyle \int \frac {e^{\frac {1}{\log \left (e^x \log \left (x^2\right )+x\right )-x-4 \log (3)}} \left (2 x^4+2 x^3+16 x^3 \log (3)+6 x^2+32 x^2 \log ^2(3)+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (e^x \log \left (x^2\right )+x\right )+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (e^x \log \left (x^2\right )+x\right )-8 x+e^x (-4 x-16)\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (e^x \log \left (x^2\right )+x\right )+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (e^x \log \left (x^2\right )+x\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{\frac {1}{\log \left (e^x \log \left (x^2\right )+x\right )-x-4 \log (3)}} \left (2 x^4+2 x^3+16 x^3 \log (3)+x^2 \left (6+32 \log ^2(3)\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (e^x \log \left (x^2\right )+x\right )+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (e^x \log \left (x^2\right )+x\right )-8 x+e^x (-4 x-16)\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (e^x \log \left (x^2\right )+x\right )+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (e^x \log \left (x^2\right )+x\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{\frac {1}{\log \left (e^x \log \left (x^2\right )+x\right )-x-4 \log (3)}} \left (2 x^4+x^3 (2+16 \log (3))+x^2 \left (6+32 \log ^2(3)\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (e^x \log \left (x^2\right )+x\right )+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (e^x \log \left (x^2\right )+x\right )-8 x+e^x (-4 x-16)\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (e^x \log \left (x^2\right )+x\right )+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (e^x \log \left (x^2\right )+x\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 e^{\frac {1}{\log \left (e^x \log \left (x^2\right )+x\right )-x-4 \log (3)}} \left (x^4+x^3 (1+8 \log (3))+x^2 \log ^2\left (e^x \log \left (x^2\right )+x\right )+3 x^2 \left (1+\frac {16 \log ^2(3)}{3}\right )-2 x^2 (x+\log (81)) \log \left (e^x \log \left (x^2\right )+x\right )+e^x x \log \left (x^2\right ) \left (-\log \left (e^x \log \left (x^2\right )+x\right )+x+\log (81)\right )^2-2 e^x x-4 x-8 e^x\right )}{x \left (e^x \log \left (x^2\right )+x\right ) \left (-\log \left (e^x \log \left (x^2\right )+x\right )+x+\log (81)\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (-x^4-(1+8 \log (3)) x^3-\log ^2\left (x+e^x \log \left (x^2\right )\right ) x^2+2 (x+\log (81)) \log \left (x+e^x \log \left (x^2\right )\right ) x^2-\left (3+16 \log ^2(3)\right ) x^2+2 e^x x-e^x \log \left (x^2\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2 x+4 x+8 e^x\right )}{x \left (x+e^x \log \left (x^2\right )\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (-x^4-(1+8 \log (3)) x^3-\log ^2\left (x+e^x \log \left (x^2\right )\right ) x^2+2 (x+\log (81)) \log \left (x+e^x \log \left (x^2\right )\right ) x^2-\left (3+16 \log ^2(3)\right ) x^2+2 e^x x-e^x \log \left (x^2\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2 x+4 x+8 e^x\right )}{x \left (x+e^x \log \left (x^2\right )\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (-\log \left (x^2\right ) x^2-3 \left (1+\frac {1}{3} \left (16 \log ^2(3)-\log ^2(81)\right )\right ) \log \left (x^2\right ) x-2 x+4 \log \left (x^2\right )-8\right )}{\log \left (x^2\right ) \left (x+e^x \log \left (x^2\right )\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}+\frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (-\log \left (x^2\right ) x^3-2 \log (81) \log \left (x^2\right ) x^2+2 \log \left (x^2\right ) \log \left (x+e^x \log \left (x^2\right )\right ) x^2-\log \left (x^2\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right ) x-\log ^2(81) \log \left (x^2\right ) x+2 \log (81) \log \left (x^2\right ) \log \left (x+e^x \log \left (x^2\right )\right ) x+2 x+8\right )}{x \log \left (x^2\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (-\log \left (x^2\right ) x^2-3 \left (1+\frac {1}{3} \left (16 \log ^2(3)-\log ^2(81)\right )\right ) \log \left (x^2\right ) x-2 x+4 \log \left (x^2\right )-8\right )}{\log \left (x^2\right ) \left (x+e^x \log \left (x^2\right )\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}-\frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (\log \left (x^2\right ) x^3+2 \log (81) \log \left (x^2\right ) x^2-2 \log \left (x^2\right ) \log \left (x+e^x \log \left (x^2\right )\right ) x^2+\log \left (x^2\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right ) x+\log ^2(81) \log \left (x^2\right ) x-2 \log (81) \log \left (x^2\right ) \log \left (x+e^x \log \left (x^2\right )\right ) x-2 x-8\right )}{x \log \left (x^2\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle -2 \int \left (\frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (-\log \left (x^2\right ) x^2-3 \left (1+\frac {1}{3} \left (16 \log ^2(3)-\log ^2(81)\right )\right ) \log \left (x^2\right ) x-2 x+4 \log \left (x^2\right )-8\right )}{\log \left (x^2\right ) \left (x+e^x \log \left (x^2\right )\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}-\frac {e^{\frac {1}{-x+\log \left (x+e^x \log \left (x^2\right )\right )-4 \log (3)}} \left (\log \left (x^2\right ) x^3+2 \log (81) \log \left (x^2\right ) x^2-2 \log \left (x^2\right ) \log \left (x+e^x \log \left (x^2\right )\right ) x^2+\log \left (x^2\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right ) x+\log ^2(81) \log \left (x^2\right ) x-2 \log (81) \log \left (x^2\right ) \log \left (x+e^x \log \left (x^2\right )\right ) x-2 x-8\right )}{x \log \left (x^2\right ) \left (x-\log \left (x+e^x \log \left (x^2\right )\right )+\log (81)\right )^2}\right )dx\) |
Int[(E^(-x - 4*Log[3] + Log[x + E^x*Log[x^2]])^(-1)*(E^x*(-16 - 4*x) - 8*x + 6*x^2 + 2*x^3 + 2*x^4 + 16*x^3*Log[3] + 32*x^2*Log[3]^2 + E^x*(2*x^3 + 16*x^2*Log[3] + 32*x*Log[3]^2)*Log[x^2] + (-4*x^3 - 16*x^2*Log[3] + E^x*(- 4*x^2 - 16*x*Log[3])*Log[x^2])*Log[x + E^x*Log[x^2]] + (2*x^2 + 2*E^x*x*Lo g[x^2])*Log[x + E^x*Log[x^2]]^2))/(x^4 + 8*x^3*Log[3] + 16*x^2*Log[3]^2 + E^x*(x^3 + 8*x^2*Log[3] + 16*x*Log[3]^2)*Log[x^2] + (-2*x^3 - 8*x^2*Log[3] + E^x*(-2*x^2 - 8*x*Log[3])*Log[x^2])*Log[x + E^x*Log[x^2]] + (x^2 + E^x* x*Log[x^2])*Log[x + E^x*Log[x^2]]^2),x]
3.17.59.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11
\[\left (2 x +8\right ) {\mathrm e}^{-\frac {1}{-\ln \left ({\mathrm e}^{x} \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x \right )+4 \ln \left (3\right )+x}}\]
int(((2*x*exp(x)*ln(x^2)+2*x^2)*ln(exp(x)*ln(x^2)+x)^2+((-16*x*ln(3)-4*x^2 )*exp(x)*ln(x^2)-16*x^2*ln(3)-4*x^3)*ln(exp(x)*ln(x^2)+x)+(32*x*ln(3)^2+16 *x^2*ln(3)+2*x^3)*exp(x)*ln(x^2)+(-16-4*x)*exp(x)+32*x^2*ln(3)^2+16*x^3*ln (3)+2*x^4+2*x^3+6*x^2-8*x)*exp(1/(ln(exp(x)*ln(x^2)+x)-4*ln(3)-x))/((x*exp (x)*ln(x^2)+x^2)*ln(exp(x)*ln(x^2)+x)^2+((-8*x*ln(3)-2*x^2)*exp(x)*ln(x^2) -8*x^2*ln(3)-2*x^3)*ln(exp(x)*ln(x^2)+x)+(16*x*ln(3)^2+8*x^2*ln(3)+x^3)*ex p(x)*ln(x^2)+16*x^2*ln(3)^2+8*x^3*ln(3)+x^4),x)
(2*x+8)*exp(-1/(-ln(exp(x)*(2*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csg n(I*x))^2)+x)+4*ln(3)+x))
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx=2 \, {\left (x + 4\right )} e^{\left (-\frac {1}{x + 4 \, \log \left (3\right ) - \log \left (e^{x} \log \left (x^{2}\right ) + x\right )}\right )} \]
integrate(((2*x*exp(x)*log(x^2)+2*x^2)*log(exp(x)*log(x^2)+x)^2+((-16*x*lo g(3)-4*x^2)*exp(x)*log(x^2)-16*x^2*log(3)-4*x^3)*log(exp(x)*log(x^2)+x)+(3 2*x*log(3)^2+16*x^2*log(3)+2*x^3)*exp(x)*log(x^2)+(-16-4*x)*exp(x)+32*x^2* log(3)^2+16*x^3*log(3)+2*x^4+2*x^3+6*x^2-8*x)*exp(1/(log(exp(x)*log(x^2)+x )-4*log(3)-x))/((x*exp(x)*log(x^2)+x^2)*log(exp(x)*log(x^2)+x)^2+((-8*x*lo g(3)-2*x^2)*exp(x)*log(x^2)-8*x^2*log(3)-2*x^3)*log(exp(x)*log(x^2)+x)+(16 *x*log(3)^2+8*x^2*log(3)+x^3)*exp(x)*log(x^2)+16*x^2*log(3)^2+8*x^3*log(3) +x^4),x, algorithm=\
Timed out. \[ \int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx=\text {Timed out} \]
integrate(((2*x*exp(x)*ln(x**2)+2*x**2)*ln(exp(x)*ln(x**2)+x)**2+((-16*x*l n(3)-4*x**2)*exp(x)*ln(x**2)-16*x**2*ln(3)-4*x**3)*ln(exp(x)*ln(x**2)+x)+( 32*x*ln(3)**2+16*x**2*ln(3)+2*x**3)*exp(x)*ln(x**2)+(-16-4*x)*exp(x)+32*x* *2*ln(3)**2+16*x**3*ln(3)+2*x**4+2*x**3+6*x**2-8*x)*exp(1/(ln(exp(x)*ln(x* *2)+x)-4*ln(3)-x))/((x*exp(x)*ln(x**2)+x**2)*ln(exp(x)*ln(x**2)+x)**2+((-8 *x*ln(3)-2*x**2)*exp(x)*ln(x**2)-8*x**2*ln(3)-2*x**3)*ln(exp(x)*ln(x**2)+x )+(16*x*ln(3)**2+8*x**2*ln(3)+x**3)*exp(x)*ln(x**2)+16*x**2*ln(3)**2+8*x** 3*ln(3)+x**4),x)
\[ \int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx=\int { \frac {2 \, {\left (x^{4} + 8 \, x^{3} \log \left (3\right ) + 16 \, x^{2} \log \left (3\right )^{2} + x^{3} + {\left (x^{3} + 8 \, x^{2} \log \left (3\right ) + 16 \, x \log \left (3\right )^{2}\right )} e^{x} \log \left (x^{2}\right ) + {\left (x e^{x} \log \left (x^{2}\right ) + x^{2}\right )} \log \left (e^{x} \log \left (x^{2}\right ) + x\right )^{2} + 3 \, x^{2} - 2 \, {\left (x + 4\right )} e^{x} - 2 \, {\left (x^{3} + 4 \, x^{2} \log \left (3\right ) + {\left (x^{2} + 4 \, x \log \left (3\right )\right )} e^{x} \log \left (x^{2}\right )\right )} \log \left (e^{x} \log \left (x^{2}\right ) + x\right ) - 4 \, x\right )} e^{\left (-\frac {1}{x + 4 \, \log \left (3\right ) - \log \left (e^{x} \log \left (x^{2}\right ) + x\right )}\right )}}{x^{4} + 8 \, x^{3} \log \left (3\right ) + 16 \, x^{2} \log \left (3\right )^{2} + {\left (x^{3} + 8 \, x^{2} \log \left (3\right ) + 16 \, x \log \left (3\right )^{2}\right )} e^{x} \log \left (x^{2}\right ) + {\left (x e^{x} \log \left (x^{2}\right ) + x^{2}\right )} \log \left (e^{x} \log \left (x^{2}\right ) + x\right )^{2} - 2 \, {\left (x^{3} + 4 \, x^{2} \log \left (3\right ) + {\left (x^{2} + 4 \, x \log \left (3\right )\right )} e^{x} \log \left (x^{2}\right )\right )} \log \left (e^{x} \log \left (x^{2}\right ) + x\right )} \,d x } \]
integrate(((2*x*exp(x)*log(x^2)+2*x^2)*log(exp(x)*log(x^2)+x)^2+((-16*x*lo g(3)-4*x^2)*exp(x)*log(x^2)-16*x^2*log(3)-4*x^3)*log(exp(x)*log(x^2)+x)+(3 2*x*log(3)^2+16*x^2*log(3)+2*x^3)*exp(x)*log(x^2)+(-16-4*x)*exp(x)+32*x^2* log(3)^2+16*x^3*log(3)+2*x^4+2*x^3+6*x^2-8*x)*exp(1/(log(exp(x)*log(x^2)+x )-4*log(3)-x))/((x*exp(x)*log(x^2)+x^2)*log(exp(x)*log(x^2)+x)^2+((-8*x*lo g(3)-2*x^2)*exp(x)*log(x^2)-8*x^2*log(3)-2*x^3)*log(exp(x)*log(x^2)+x)+(16 *x*log(3)^2+8*x^2*log(3)+x^3)*exp(x)*log(x^2)+16*x^2*log(3)^2+8*x^3*log(3) +x^4),x, algorithm=\
2*x^4*e^(-1/(x + 4*log(3) - log(2*e^x*log(x) + x)))/(x^2 - x - 2*e^x) + 16 *x^3*e^(-1/(x + 4*log(3) - log(2*e^x*log(x) + x)))*log(3)/(x^2 - x - 2*e^x ) + 32*x^2*e^(-1/(x + 4*log(3) - log(2*e^x*log(x) + x)))*log(3)^2/(x^2 - x - 2*e^x) + 4*x^3*e^(x - 1/(x + 4*log(3) - log(2*e^x*log(x) + x)))*log(x)/ (x^2 - x - 2*e^x) + 32*x^2*e^(x - 1/(x + 4*log(3) - log(2*e^x*log(x) + x)) )*log(3)*log(x)/(x^2 - x - 2*e^x) + 64*x*e^(x - 1/(x + 4*log(3) - log(2*e^ x*log(x) + x)))*log(3)^2*log(x)/(x^2 - x - 2*e^x) + 2*x^3*e^(-1/(x + 4*log (3) - log(2*e^x*log(x) + x)))/(x^2 - x - 2*e^x) + 6*x^2*e^(-1/(x + 4*log(3 ) - log(2*e^x*log(x) + x)))/(x^2 - x - 2*e^x) - 4*x*e^(x - 1/(x + 4*log(3) - log(2*e^x*log(x) + x)))/(x^2 - x - 2*e^x) - 8*x*e^(-1/(x + 4*log(3) - l og(2*e^x*log(x) + x)))/(x^2 - x - 2*e^x) - 16*e^(x - 1/(x + 4*log(3) - log (2*e^x*log(x) + x)))/(x^2 - x - 2*e^x) + 2*integrate(-(2*(x + 4*log(3))*lo g(2*e^x*log(x) + x) - log(2*e^x*log(x) + x)^2)*e^(-1/(x + 4*log(3) - log(2 *e^x*log(x) + x)))/(x^2 + 8*x*log(3) + 16*log(3)^2 - 2*(x + 4*log(3))*log( 2*e^x*log(x) + x) + log(2*e^x*log(x) + x)^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (28) = 56\).
Time = 3.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.86 \[ \int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx=2 \, x e^{\left (\frac {x - \log \left (e^{x} \log \left (x^{2}\right ) + x\right )}{4 \, {\left (x \log \left (3\right ) + 4 \, \log \left (3\right )^{2} - \log \left (3\right ) \log \left (e^{x} \log \left (x^{2}\right ) + x\right )\right )}} - \frac {1}{4 \, \log \left (3\right )}\right )} + 8 \, e^{\left (\frac {x - \log \left (e^{x} \log \left (x^{2}\right ) + x\right )}{4 \, {\left (x \log \left (3\right ) + 4 \, \log \left (3\right )^{2} - \log \left (3\right ) \log \left (e^{x} \log \left (x^{2}\right ) + x\right )\right )}} - \frac {1}{4 \, \log \left (3\right )}\right )} \]
integrate(((2*x*exp(x)*log(x^2)+2*x^2)*log(exp(x)*log(x^2)+x)^2+((-16*x*lo g(3)-4*x^2)*exp(x)*log(x^2)-16*x^2*log(3)-4*x^3)*log(exp(x)*log(x^2)+x)+(3 2*x*log(3)^2+16*x^2*log(3)+2*x^3)*exp(x)*log(x^2)+(-16-4*x)*exp(x)+32*x^2* log(3)^2+16*x^3*log(3)+2*x^4+2*x^3+6*x^2-8*x)*exp(1/(log(exp(x)*log(x^2)+x )-4*log(3)-x))/((x*exp(x)*log(x^2)+x^2)*log(exp(x)*log(x^2)+x)^2+((-8*x*lo g(3)-2*x^2)*exp(x)*log(x^2)-8*x^2*log(3)-2*x^3)*log(exp(x)*log(x^2)+x)+(16 *x*log(3)^2+8*x^2*log(3)+x^3)*exp(x)*log(x^2)+16*x^2*log(3)^2+8*x^3*log(3) +x^4),x, algorithm=\
2*x*e^(1/4*(x - log(e^x*log(x^2) + x))/(x*log(3) + 4*log(3)^2 - log(3)*log (e^x*log(x^2) + x)) - 1/4/log(3)) + 8*e^(1/4*(x - log(e^x*log(x^2) + x))/( x*log(3) + 4*log(3)^2 - log(3)*log(e^x*log(x^2) + x)) - 1/4/log(3))
Timed out. \[ \int \frac {e^{\frac {1}{-x-4 \log (3)+\log \left (x+e^x \log \left (x^2\right )\right )}} \left (e^x (-16-4 x)-8 x+6 x^2+2 x^3+2 x^4+16 x^3 \log (3)+32 x^2 \log ^2(3)+e^x \left (2 x^3+16 x^2 \log (3)+32 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-4 x^3-16 x^2 \log (3)+e^x \left (-4 x^2-16 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (2 x^2+2 e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )\right )}{x^4+8 x^3 \log (3)+16 x^2 \log ^2(3)+e^x \left (x^3+8 x^2 \log (3)+16 x \log ^2(3)\right ) \log \left (x^2\right )+\left (-2 x^3-8 x^2 \log (3)+e^x \left (-2 x^2-8 x \log (3)\right ) \log \left (x^2\right )\right ) \log \left (x+e^x \log \left (x^2\right )\right )+\left (x^2+e^x x \log \left (x^2\right )\right ) \log ^2\left (x+e^x \log \left (x^2\right )\right )} \, dx=\int \frac {{\mathrm {e}}^{-\frac {1}{x+4\,\ln \left (3\right )-\ln \left (x+\ln \left (x^2\right )\,{\mathrm {e}}^x\right )}}\,\left (32\,x^2\,{\ln \left (3\right )}^2-8\,x+{\ln \left (x+\ln \left (x^2\right )\,{\mathrm {e}}^x\right )}^2\,\left (2\,x^2+2\,x\,\ln \left (x^2\right )\,{\mathrm {e}}^x\right )-{\mathrm {e}}^x\,\left (4\,x+16\right )+16\,x^3\,\ln \left (3\right )-\ln \left (x+\ln \left (x^2\right )\,{\mathrm {e}}^x\right )\,\left (16\,x^2\,\ln \left (3\right )+4\,x^3+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (4\,x^2+16\,\ln \left (3\right )\,x\right )\right )+6\,x^2+2\,x^3+2\,x^4+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (2\,x^3+16\,\ln \left (3\right )\,x^2+32\,{\ln \left (3\right )}^2\,x\right )\right )}{16\,x^2\,{\ln \left (3\right )}^2+8\,x^3\,\ln \left (3\right )+{\ln \left (x+\ln \left (x^2\right )\,{\mathrm {e}}^x\right )}^2\,\left (x^2+x\,\ln \left (x^2\right )\,{\mathrm {e}}^x\right )-\ln \left (x+\ln \left (x^2\right )\,{\mathrm {e}}^x\right )\,\left (8\,x^2\,\ln \left (3\right )+2\,x^3+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (2\,x^2+8\,\ln \left (3\right )\,x\right )\right )+x^4+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (x^3+8\,\ln \left (3\right )\,x^2+16\,{\ln \left (3\right )}^2\,x\right )} \,d x \]
int((exp(-1/(x + 4*log(3) - log(x + log(x^2)*exp(x))))*(32*x^2*log(3)^2 - 8*x + log(x + log(x^2)*exp(x))^2*(2*x^2 + 2*x*log(x^2)*exp(x)) - exp(x)*(4 *x + 16) + 16*x^3*log(3) - log(x + log(x^2)*exp(x))*(16*x^2*log(3) + 4*x^3 + log(x^2)*exp(x)*(16*x*log(3) + 4*x^2)) + 6*x^2 + 2*x^3 + 2*x^4 + log(x^ 2)*exp(x)*(32*x*log(3)^2 + 16*x^2*log(3) + 2*x^3)))/(16*x^2*log(3)^2 + 8*x ^3*log(3) + log(x + log(x^2)*exp(x))^2*(x^2 + x*log(x^2)*exp(x)) - log(x + log(x^2)*exp(x))*(8*x^2*log(3) + 2*x^3 + log(x^2)*exp(x)*(8*x*log(3) + 2* x^2)) + x^4 + log(x^2)*exp(x)*(16*x*log(3)^2 + 8*x^2*log(3) + x^3)),x)
int((exp(-1/(x + 4*log(3) - log(x + log(x^2)*exp(x))))*(32*x^2*log(3)^2 - 8*x + log(x + log(x^2)*exp(x))^2*(2*x^2 + 2*x*log(x^2)*exp(x)) - exp(x)*(4 *x + 16) + 16*x^3*log(3) - log(x + log(x^2)*exp(x))*(16*x^2*log(3) + 4*x^3 + log(x^2)*exp(x)*(16*x*log(3) + 4*x^2)) + 6*x^2 + 2*x^3 + 2*x^4 + log(x^ 2)*exp(x)*(32*x*log(3)^2 + 16*x^2*log(3) + 2*x^3)))/(16*x^2*log(3)^2 + 8*x ^3*log(3) + log(x + log(x^2)*exp(x))^2*(x^2 + x*log(x^2)*exp(x)) - log(x + log(x^2)*exp(x))*(8*x^2*log(3) + 2*x^3 + log(x^2)*exp(x)*(8*x*log(3) + 2* x^2)) + x^4 + log(x^2)*exp(x)*(16*x*log(3)^2 + 8*x^2*log(3) + x^3)), x)