\[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{x}}{\ln \! \left (\ln \! \left (x +\frac {\left (2+\frac {3}{x^{2}-2 x -1}\right ) \left (\frac {2}{3}+\frac {1}{x^{2}-2 x -1}\right )}{3}\right )-x \right )} \]
command
Int[(E^x*(-4 + 111*x - 48*x^2 - 125*x^3 + 3*x^4 + 111*x^5 - 59*x^6 + 9*x^7) + (E^x*(x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8) + E^x*(-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9*x^7)*Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]])/((x + 3*x^2 + 57*x^3 + 113*x^4 - 84*x^5 - 57*x^6 + 50*x^7 - 9*x^8 + (-1 - 3*x - 57*x^2 - 113*x^3 + 84*x^4 + 57*x^5 - 50*x^6 + 9*x^7)*Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)])*Log[-x + Log[(1 + x + 56*x^2 + 2*x^3 - 32*x^4 + 9*x^5)/(9 + 36*x + 18*x^2 - 36*x^3 + 9*x^4)]]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^x}{\log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (-x^2+2 x+1\right )^2}\right )-x\right )} \]