\[ \int \frac {\left (8 e^{10} x+4 x^2+14 x^3+16 x^4+8 x^5+e^5 \left (2 x+16 x^2+16 x^3\right )\right ) \log \left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )+\left (-2 x^2-10 x^3-14 x^4-4 x^5+2 x^6+e^{10} \left (-8 x+2 x^2\right )+e^5 \left (-2 x-16 x^2-12 x^3+4 x^4\right )\right ) \log ^2\left (\frac {1+e^5 (4-x)+4 x+3 x^2-x^3}{e^5 x+x^2+x^3}\right )}{e^{10} (-4+x)-x-5 x^2-7 x^3-2 x^4+x^5+e^5 \left (-1-8 x-6 x^2+2 x^3\right )} \, dx \]
Optimal antiderivative \[ x^{2} {\ln \! \left (\frac {4+\frac {1}{x^{2}+{\mathrm e}^{5}+x}-x}{x}\right )}^{2} \]
command
Int[((8*E^10*x + 4*x^2 + 14*x^3 + 16*x^4 + 8*x^5 + E^5*(2*x + 16*x^2 + 16*x^3))*Log[(1 + E^5*(4 - x) + 4*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)] + (-2*x^2 - 10*x^3 - 14*x^4 - 4*x^5 + 2*x^6 + E^10*(-8*x + 2*x^2) + E^5*(-2*x - 16*x^2 - 12*x^3 + 4*x^4))*Log[(1 + E^5*(4 - x) + 4*x + 3*x^2 - x^3)/(E^5*x + x^2 + x^3)]^2)/(E^10*(-4 + x) - x - 5*x^2 - 7*x^3 - 2*x^4 + x^5 + E^5*(-1 - 8*x - 6*x^2 + 2*x^3)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ x^2 \log ^2\left (\frac {-x^3+3 x^2+\left (4-e^5\right ) x+4 e^5+1}{x \left (x^2+x+e^5\right )}\right ) \]