Integrand size = 336, antiderivative size = 25 \[ \int \frac {e^x \left (10 x+14 x^2+4 x^3+\left (16 x+8 x^2\right ) \log (3)+4 x \log ^2(3)\right )+\left (e^x \left (2 x+5 x^2+5 x^3+2 x^4+\left (4 x+8 x^2+4 x^3\right ) \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)\right )+e^x \left (2+5 x+5 x^2+2 x^3+\left (4+8 x+4 x^2\right ) \log (3)+(2+2 x) \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (\log ^2\left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )\right )}{\left (2 x+3 x^2+2 x^3+\left (4 x+4 x^2\right ) \log (3)+2 x \log ^2(3)+\left (2+3 x+2 x^2+(4+4 x) \log (3)+2 \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )} \, dx=-3+e^x x \log \left (\log ^2\left (x+\log \left (x-2 (1+x+\log (3))^2\right )\right )\right ) \]
Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^x \left (10 x+14 x^2+4 x^3+\left (16 x+8 x^2\right ) \log (3)+4 x \log ^2(3)\right )+\left (e^x \left (2 x+5 x^2+5 x^3+2 x^4+\left (4 x+8 x^2+4 x^3\right ) \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)\right )+e^x \left (2+5 x+5 x^2+2 x^3+\left (4+8 x+4 x^2\right ) \log (3)+(2+2 x) \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (\log ^2\left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )\right )}{\left (2 x+3 x^2+2 x^3+\left (4 x+4 x^2\right ) \log (3)+2 x \log ^2(3)+\left (2+3 x+2 x^2+(4+4 x) \log (3)+2 \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )} \, dx=e^x x \log \left (\log ^2\left (x+\log \left (-2 x^2-2 (1+\log (3))^2-x (3+\log (81))\right )\right )\right ) \]
Integrate[(E^x*(10*x + 14*x^2 + 4*x^3 + (16*x + 8*x^2)*Log[3] + 4*x*Log[3] ^2) + (E^x*(2*x + 5*x^2 + 5*x^3 + 2*x^4 + (4*x + 8*x^2 + 4*x^3)*Log[3] + ( 2*x + 2*x^2)*Log[3]^2) + E^x*(2 + 5*x + 5*x^2 + 2*x^3 + (4 + 8*x + 4*x^2)* Log[3] + (2 + 2*x)*Log[3]^2)*Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2* Log[3]^2])*Log[x + Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Log[3]^2]] *Log[Log[x + Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Log[3]^2]]^2])/( (2*x + 3*x^2 + 2*x^3 + (4*x + 4*x^2)*Log[3] + 2*x*Log[3]^2 + (2 + 3*x + 2* x^2 + (4 + 4*x)*Log[3] + 2*Log[3]^2)*Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log [3] - 2*Log[3]^2])*Log[x + Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Lo g[3]^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.
| \(\displaystyle \int \frac {e^x \left (4 x^3+14 x^2+\left (8 x^2+16 x\right ) \log (3)+10 x+4 x \log ^2(3)\right )+\left (e^x \left (2 x^3+5 x^2+\left (4 x^2+8 x+4\right ) \log (3)+5 x+(2 x+2) \log ^2(3)+2\right ) \log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+e^x \left (2 x^4+5 x^3+5 x^2+\left (2 x^2+2 x\right ) \log ^2(3)+\left (4 x^3+8 x^2+4 x\right ) \log (3)+2 x\right )\right ) \log \left (\log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+x\right ) \log \left (\log ^2\left (\log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+x\right )\right )}{\left (2 x^3+3 x^2+\left (2 x^2+3 x+(4 x+4) \log (3)+2+2 \log ^2(3)\right ) \log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+\left (4 x^2+4 x\right ) \log (3)+2 x+2 x \log ^2(3)\right ) \log \left (\log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+x\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^x \left (4 x^3+14 x^2+\left (8 x^2+16 x\right ) \log (3)+10 x+4 x \log ^2(3)\right )+\left (e^x \left (2 x^3+5 x^2+\left (4 x^2+8 x+4\right ) \log (3)+5 x+(2 x+2) \log ^2(3)+2\right ) \log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+e^x \left (2 x^4+5 x^3+5 x^2+\left (2 x^2+2 x\right ) \log ^2(3)+\left (4 x^3+8 x^2+4 x\right ) \log (3)+2 x\right )\right ) \log \left (\log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+x\right ) \log \left (\log ^2\left (\log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+x\right )\right )}{\left (2 x^3+3 x^2+\left (2 x^2+3 x+(4 x+4) \log (3)+2+2 \log ^2(3)\right ) \log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+\left (4 x^2+4 x\right ) \log (3)+x \left (2+2 \log ^2(3)\right )\right ) \log \left (\log \left (-2 x^2-3 x+(-4 x-4) \log (3)-2-2 \log ^2(3)\right )+x\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int e^x \left (\frac {2 x \left (2 x^2+x (7+\log (81))+5+2 \log ^2(3)+8 \log (3)\right )}{\left (2 x^2+x (3+\log (81))+2+2 \log ^2(3)+\log (81)\right ) \left (\log \left (-2 x^2-x (3+\log (81))-2 (1+\log (3))^2\right )+x\right ) \log \left (\log \left (-2 x^2-x (3+\log (81))-2 (1+\log (3))^2\right )+x\right )}+(x+1) \log \left (\log ^2\left (\log \left (-2 x^2-x (3+\log (81))-2 (1+\log (3))^2\right )+x\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^x x \left (2 x^2+x (7+\log (81))+5+2 \log ^2(3)+8 \log (3)\right )}{\left (2 x^2+x (3+\log (81))+2+2 \log ^2(3)+\log (81)\right ) \left (\log \left (-2 x^2-x (3+\log (81))-2 (1+\log (3))^2\right )+x\right ) \log \left (\log \left (-2 x^2-x (3+\log (81))-2 (1+\log (3))^2\right )+x\right )}+e^x (x+1) \log \left (\log ^2\left (\log \left (-2 x^2-x (3+\log (81))-2 (1+\log (3))^2\right )+x\right )\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 (3+\log (81)) \left (1+\frac {i (3+\log (81))}{\sqrt {7+16 \log ^2(3)-\log ^2(81)+\log (6561)}}\right ) \int \frac {e^x}{\left (4 x-i \sqrt {7+16 \log ^2(3)+2 \log (81)-\log ^2(81)}+\log (81)+3\right ) \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right ) \log \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )}dx-\frac {32 i \left (1+\log ^2(3)+\log (9)\right ) \int \frac {e^x}{\left (-4 x+i \sqrt {7+16 \log ^2(3)+2 \log (81)-\log ^2(81)}-\log (81)-3\right ) \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right ) \log \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )}dx}{\sqrt {7+16 \log ^2(3)-\log ^2(81)+\log (6561)}}-2 (3+\log (81)) \left (1-\frac {i (3+\log (81))}{\sqrt {7+16 \log ^2(3)-\log ^2(81)+\log (6561)}}\right ) \int \frac {e^x}{\left (4 x+i \sqrt {7+16 \log ^2(3)+2 \log (81)-\log ^2(81)}+\log (81)+3\right ) \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right ) \log \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )}dx-\frac {32 i \left (1+\log ^2(3)+\log (9)\right ) \int \frac {e^x}{\left (4 x+i \sqrt {7+16 \log ^2(3)+2 \log (81)-\log ^2(81)}+\log (81)+3\right ) \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right ) \log \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )}dx}{\sqrt {7+16 \log ^2(3)-\log ^2(81)+\log (6561)}}+\int e^x \log \left (\log ^2\left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )\right )dx+\int e^x x \log \left (\log ^2\left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )\right )dx+4 \int \frac {e^x}{\left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right ) \log \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )}dx+2 \int \frac {e^x x}{\left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right ) \log \left (x+\log \left (-2 x^2-(3+\log (81)) x-2 (1+\log (3))^2\right )\right )}dx\) |
Int[(E^x*(10*x + 14*x^2 + 4*x^3 + (16*x + 8*x^2)*Log[3] + 4*x*Log[3]^2) + (E^x*(2*x + 5*x^2 + 5*x^3 + 2*x^4 + (4*x + 8*x^2 + 4*x^3)*Log[3] + (2*x + 2*x^2)*Log[3]^2) + E^x*(2 + 5*x + 5*x^2 + 2*x^3 + (4 + 8*x + 4*x^2)*Log[3] + (2 + 2*x)*Log[3]^2)*Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Log[3] ^2])*Log[x + Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Log[3]^2]]*Log[L og[x + Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Log[3]^2]]^2])/((2*x + 3*x^2 + 2*x^3 + (4*x + 4*x^2)*Log[3] + 2*x*Log[3]^2 + (2 + 3*x + 2*x^2 + (4 + 4*x)*Log[3] + 2*Log[3]^2)*Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Log[3]^2])*Log[x + Log[-2 - 3*x - 2*x^2 + (-4 - 4*x)*Log[3] - 2*Log[3]^2 ]]),x]
3.9.2.3.1 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[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 = 0.42 (sec) , antiderivative size = 216, normalized size of antiderivative = 8.64
\[2 x \,{\mathrm e}^{x} \ln \left (\ln \left (\ln \left (-2 \ln \left (3\right )^{2}+\left (-4-4 x \right ) \ln \left (3\right )-2 x^{2}-3 x -2\right )+x \right )\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\ln \left (\ln \left (-2 \ln \left (3\right )^{2}+\left (-4-4 x \right ) \ln \left (3\right )-2 x^{2}-3 x -2\right )+x \right )}^{2}\right ) \left ({\operatorname {csgn}\left (i \ln \left (\ln \left (-2 \ln \left (3\right )^{2}+\left (-4-4 x \right ) \ln \left (3\right )-2 x^{2}-3 x -2\right )+x \right )\right )}^{2}-2 \,\operatorname {csgn}\left (i {\ln \left (\ln \left (-2 \ln \left (3\right )^{2}+\left (-4-4 x \right ) \ln \left (3\right )-2 x^{2}-3 x -2\right )+x \right )}^{2}\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (-2 \ln \left (3\right )^{2}+\left (-4-4 x \right ) \ln \left (3\right )-2 x^{2}-3 x -2\right )+x \right )\right )+{\operatorname {csgn}\left (i {\ln \left (\ln \left (-2 \ln \left (3\right )^{2}+\left (-4-4 x \right ) \ln \left (3\right )-2 x^{2}-3 x -2\right )+x \right )}^{2}\right )}^{2}\right ) x \,{\mathrm e}^{x}}{2}\]
int(((((2+2*x)*ln(3)^2+(4*x^2+8*x+4)*ln(3)+2*x^3+5*x^2+5*x+2)*exp(x)*ln(-2 *ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x-2)+((2*x^2+2*x)*ln(3)^2+(4*x^3+8*x^2+4*x )*ln(3)+2*x^4+5*x^3+5*x^2+2*x)*exp(x))*ln(ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x ^2-3*x-2)+x)*ln(ln(ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x-2)+x)^2)+(4*x*ln (3)^2+(8*x^2+16*x)*ln(3)+4*x^3+14*x^2+10*x)*exp(x))/((2*ln(3)^2+(4+4*x)*ln (3)+2*x^2+3*x+2)*ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x-2)+2*x*ln(3)^2+(4* x^2+4*x)*ln(3)+2*x^3+3*x^2+2*x)/ln(ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x- 2)+x),x)
2*x*exp(x)*ln(ln(ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x-2)+x))-1/2*I*Pi*cs gn(I*ln(ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x-2)+x)^2)*(csgn(I*ln(ln(-2*l n(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x-2)+x))^2-2*csgn(I*ln(ln(-2*ln(3)^2+(-4-4*x )*ln(3)-2*x^2-3*x-2)+x)^2)*csgn(I*ln(ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3* x-2)+x))+csgn(I*ln(ln(-2*ln(3)^2+(-4-4*x)*ln(3)-2*x^2-3*x-2)+x)^2)^2)*x*ex p(x)
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^x \left (10 x+14 x^2+4 x^3+\left (16 x+8 x^2\right ) \log (3)+4 x \log ^2(3)\right )+\left (e^x \left (2 x+5 x^2+5 x^3+2 x^4+\left (4 x+8 x^2+4 x^3\right ) \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)\right )+e^x \left (2+5 x+5 x^2+2 x^3+\left (4+8 x+4 x^2\right ) \log (3)+(2+2 x) \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (\log ^2\left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )\right )}{\left (2 x+3 x^2+2 x^3+\left (4 x+4 x^2\right ) \log (3)+2 x \log ^2(3)+\left (2+3 x+2 x^2+(4+4 x) \log (3)+2 \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )} \, dx=x e^{x} \log \left (\log \left (x + \log \left (-2 \, x^{2} - 4 \, {\left (x + 1\right )} \log \left (3\right ) - 2 \, \log \left (3\right )^{2} - 3 \, x - 2\right )\right )^{2}\right ) \]
integrate(((((2+2*x)*log(3)^2+(4*x^2+8*x+4)*log(3)+2*x^3+5*x^2+5*x+2)*exp( x)*log(-2*log(3)^2+(-4-4*x)*log(3)-2*x^2-3*x-2)+((2*x^2+2*x)*log(3)^2+(4*x ^3+8*x^2+4*x)*log(3)+2*x^4+5*x^3+5*x^2+2*x)*exp(x))*log(log(-2*log(3)^2+(- 4-4*x)*log(3)-2*x^2-3*x-2)+x)*log(log(log(-2*log(3)^2+(-4-4*x)*log(3)-2*x^ 2-3*x-2)+x)^2)+(4*x*log(3)^2+(8*x^2+16*x)*log(3)+4*x^3+14*x^2+10*x)*exp(x) )/((2*log(3)^2+(4+4*x)*log(3)+2*x^2+3*x+2)*log(-2*log(3)^2+(-4-4*x)*log(3) -2*x^2-3*x-2)+2*x*log(3)^2+(4*x^2+4*x)*log(3)+2*x^3+3*x^2+2*x)/log(log(-2* log(3)^2+(-4-4*x)*log(3)-2*x^2-3*x-2)+x),x, algorithm=\
Timed out. \[ \int \frac {e^x \left (10 x+14 x^2+4 x^3+\left (16 x+8 x^2\right ) \log (3)+4 x \log ^2(3)\right )+\left (e^x \left (2 x+5 x^2+5 x^3+2 x^4+\left (4 x+8 x^2+4 x^3\right ) \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)\right )+e^x \left (2+5 x+5 x^2+2 x^3+\left (4+8 x+4 x^2\right ) \log (3)+(2+2 x) \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (\log ^2\left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )\right )}{\left (2 x+3 x^2+2 x^3+\left (4 x+4 x^2\right ) \log (3)+2 x \log ^2(3)+\left (2+3 x+2 x^2+(4+4 x) \log (3)+2 \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )} \, dx=\text {Timed out} \]
integrate(((((2+2*x)*ln(3)**2+(4*x**2+8*x+4)*ln(3)+2*x**3+5*x**2+5*x+2)*ex p(x)*ln(-2*ln(3)**2+(-4-4*x)*ln(3)-2*x**2-3*x-2)+((2*x**2+2*x)*ln(3)**2+(4 *x**3+8*x**2+4*x)*ln(3)+2*x**4+5*x**3+5*x**2+2*x)*exp(x))*ln(ln(-2*ln(3)** 2+(-4-4*x)*ln(3)-2*x**2-3*x-2)+x)*ln(ln(ln(-2*ln(3)**2+(-4-4*x)*ln(3)-2*x* *2-3*x-2)+x)**2)+(4*x*ln(3)**2+(8*x**2+16*x)*ln(3)+4*x**3+14*x**2+10*x)*ex p(x))/((2*ln(3)**2+(4+4*x)*ln(3)+2*x**2+3*x+2)*ln(-2*ln(3)**2+(-4-4*x)*ln( 3)-2*x**2-3*x-2)+2*x*ln(3)**2+(4*x**2+4*x)*ln(3)+2*x**3+3*x**2+2*x)/ln(ln( -2*ln(3)**2+(-4-4*x)*ln(3)-2*x**2-3*x-2)+x),x)
Time = 0.42 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^x \left (10 x+14 x^2+4 x^3+\left (16 x+8 x^2\right ) \log (3)+4 x \log ^2(3)\right )+\left (e^x \left (2 x+5 x^2+5 x^3+2 x^4+\left (4 x+8 x^2+4 x^3\right ) \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)\right )+e^x \left (2+5 x+5 x^2+2 x^3+\left (4+8 x+4 x^2\right ) \log (3)+(2+2 x) \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (\log ^2\left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )\right )}{\left (2 x+3 x^2+2 x^3+\left (4 x+4 x^2\right ) \log (3)+2 x \log ^2(3)+\left (2+3 x+2 x^2+(4+4 x) \log (3)+2 \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )} \, dx=2 \, x e^{x} \log \left (\log \left (x + \log \left (-2 \, x^{2} - x {\left (4 \, \log \left (3\right ) + 3\right )} - 2 \, \log \left (3\right )^{2} - 4 \, \log \left (3\right ) - 2\right )\right )\right ) \]
integrate(((((2+2*x)*log(3)^2+(4*x^2+8*x+4)*log(3)+2*x^3+5*x^2+5*x+2)*exp( x)*log(-2*log(3)^2+(-4-4*x)*log(3)-2*x^2-3*x-2)+((2*x^2+2*x)*log(3)^2+(4*x ^3+8*x^2+4*x)*log(3)+2*x^4+5*x^3+5*x^2+2*x)*exp(x))*log(log(-2*log(3)^2+(- 4-4*x)*log(3)-2*x^2-3*x-2)+x)*log(log(log(-2*log(3)^2+(-4-4*x)*log(3)-2*x^ 2-3*x-2)+x)^2)+(4*x*log(3)^2+(8*x^2+16*x)*log(3)+4*x^3+14*x^2+10*x)*exp(x) )/((2*log(3)^2+(4+4*x)*log(3)+2*x^2+3*x+2)*log(-2*log(3)^2+(-4-4*x)*log(3) -2*x^2-3*x-2)+2*x*log(3)^2+(4*x^2+4*x)*log(3)+2*x^3+3*x^2+2*x)/log(log(-2* log(3)^2+(-4-4*x)*log(3)-2*x^2-3*x-2)+x),x, algorithm=\
Time = 20.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {e^x \left (10 x+14 x^2+4 x^3+\left (16 x+8 x^2\right ) \log (3)+4 x \log ^2(3)\right )+\left (e^x \left (2 x+5 x^2+5 x^3+2 x^4+\left (4 x+8 x^2+4 x^3\right ) \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)\right )+e^x \left (2+5 x+5 x^2+2 x^3+\left (4+8 x+4 x^2\right ) \log (3)+(2+2 x) \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (\log ^2\left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )\right )}{\left (2 x+3 x^2+2 x^3+\left (4 x+4 x^2\right ) \log (3)+2 x \log ^2(3)+\left (2+3 x+2 x^2+(4+4 x) \log (3)+2 \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )} \, dx=x e^{x} \log \left (\log \left (x + \log \left (-2 \, x^{2} - 4 \, x \log \left (3\right ) - 2 \, \log \left (3\right )^{2} - 3 \, x - 4 \, \log \left (3\right ) - 2\right )\right )^{2}\right ) \]
integrate(((((2+2*x)*log(3)^2+(4*x^2+8*x+4)*log(3)+2*x^3+5*x^2+5*x+2)*exp( x)*log(-2*log(3)^2+(-4-4*x)*log(3)-2*x^2-3*x-2)+((2*x^2+2*x)*log(3)^2+(4*x ^3+8*x^2+4*x)*log(3)+2*x^4+5*x^3+5*x^2+2*x)*exp(x))*log(log(-2*log(3)^2+(- 4-4*x)*log(3)-2*x^2-3*x-2)+x)*log(log(log(-2*log(3)^2+(-4-4*x)*log(3)-2*x^ 2-3*x-2)+x)^2)+(4*x*log(3)^2+(8*x^2+16*x)*log(3)+4*x^3+14*x^2+10*x)*exp(x) )/((2*log(3)^2+(4+4*x)*log(3)+2*x^2+3*x+2)*log(-2*log(3)^2+(-4-4*x)*log(3) -2*x^2-3*x-2)+2*x*log(3)^2+(4*x^2+4*x)*log(3)+2*x^3+3*x^2+2*x)/log(log(-2* log(3)^2+(-4-4*x)*log(3)-2*x^2-3*x-2)+x),x, algorithm=\
Timed out. \[ \int \frac {e^x \left (10 x+14 x^2+4 x^3+\left (16 x+8 x^2\right ) \log (3)+4 x \log ^2(3)\right )+\left (e^x \left (2 x+5 x^2+5 x^3+2 x^4+\left (4 x+8 x^2+4 x^3\right ) \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)\right )+e^x \left (2+5 x+5 x^2+2 x^3+\left (4+8 x+4 x^2\right ) \log (3)+(2+2 x) \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (\log ^2\left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )\right )}{\left (2 x+3 x^2+2 x^3+\left (4 x+4 x^2\right ) \log (3)+2 x \log ^2(3)+\left (2+3 x+2 x^2+(4+4 x) \log (3)+2 \log ^2(3)\right ) \log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right ) \log \left (x+\log \left (-2-3 x-2 x^2+(-4-4 x) \log (3)-2 \log ^2(3)\right )\right )} \, dx=\int \frac {{\mathrm {e}}^x\,\left (10\,x+\ln \left (3\right )\,\left (8\,x^2+16\,x\right )+4\,x\,{\ln \left (3\right )}^2+14\,x^2+4\,x^3\right )+\ln \left ({\ln \left (x+\ln \left (-3\,x-\ln \left (3\right )\,\left (4\,x+4\right )-2\,{\ln \left (3\right )}^2-2\,x^2-2\right )\right )}^2\right )\,\ln \left (x+\ln \left (-3\,x-\ln \left (3\right )\,\left (4\,x+4\right )-2\,{\ln \left (3\right )}^2-2\,x^2-2\right )\right )\,\left ({\mathrm {e}}^x\,\left (2\,x+\ln \left (3\right )\,\left (4\,x^3+8\,x^2+4\,x\right )+{\ln \left (3\right )}^2\,\left (2\,x^2+2\,x\right )+5\,x^2+5\,x^3+2\,x^4\right )+\ln \left (-3\,x-\ln \left (3\right )\,\left (4\,x+4\right )-2\,{\ln \left (3\right )}^2-2\,x^2-2\right )\,{\mathrm {e}}^x\,\left (5\,x+\ln \left (3\right )\,\left (4\,x^2+8\,x+4\right )+{\ln \left (3\right )}^2\,\left (2\,x+2\right )+5\,x^2+2\,x^3+2\right )\right )}{\ln \left (x+\ln \left (-3\,x-\ln \left (3\right )\,\left (4\,x+4\right )-2\,{\ln \left (3\right )}^2-2\,x^2-2\right )\right )\,\left (2\,x+\ln \left (-3\,x-\ln \left (3\right )\,\left (4\,x+4\right )-2\,{\ln \left (3\right )}^2-2\,x^2-2\right )\,\left (3\,x+\ln \left (3\right )\,\left (4\,x+4\right )+2\,{\ln \left (3\right )}^2+2\,x^2+2\right )+\ln \left (3\right )\,\left (4\,x^2+4\,x\right )+2\,x\,{\ln \left (3\right )}^2+3\,x^2+2\,x^3\right )} \,d x \]
int((exp(x)*(10*x + log(3)*(16*x + 8*x^2) + 4*x*log(3)^2 + 14*x^2 + 4*x^3) + log(log(x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2))^2)* log(x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2))*(exp(x)*(2 *x + log(3)*(4*x + 8*x^2 + 4*x^3) + log(3)^2*(2*x + 2*x^2) + 5*x^2 + 5*x^3 + 2*x^4) + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2)*exp(x)* (5*x + log(3)*(8*x + 4*x^2 + 4) + log(3)^2*(2*x + 2) + 5*x^2 + 2*x^3 + 2)) )/(log(x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2))*(2*x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2)*(3*x + log(3)*(4*x + 4) + 2*log(3)^2 + 2*x^2 + 2) + log(3)*(4*x + 4*x^2) + 2*x*log(3)^2 + 3*x ^2 + 2*x^3)),x)
int((exp(x)*(10*x + log(3)*(16*x + 8*x^2) + 4*x*log(3)^2 + 14*x^2 + 4*x^3) + log(log(x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2))^2)* log(x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2))*(exp(x)*(2 *x + log(3)*(4*x + 8*x^2 + 4*x^3) + log(3)^2*(2*x + 2*x^2) + 5*x^2 + 5*x^3 + 2*x^4) + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2)*exp(x)* (5*x + log(3)*(8*x + 4*x^2 + 4) + log(3)^2*(2*x + 2) + 5*x^2 + 2*x^3 + 2)) )/(log(x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2))*(2*x + log(- 3*x - log(3)*(4*x + 4) - 2*log(3)^2 - 2*x^2 - 2)*(3*x + log(3)*(4*x + 4) + 2*log(3)^2 + 2*x^2 + 2) + log(3)*(4*x + 4*x^2) + 2*x*log(3)^2 + 3*x ^2 + 2*x^3)), x)