\[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {1}{2+\ln \! \left (x \left (3-{\mathrm e}^{4}+x \right )\right )}+x^{3} \]
command
Int[(3 + 2*x - 36*x^3 - 12*x^4 + E^4*(-1 + 12*x^3) + (-36*x^3 + 12*E^4*x^3 - 12*x^4)*Log[3*x - E^4*x + x^2] + (-9*x^3 + 3*E^4*x^3 - 3*x^4)*Log[3*x - E^4*x + x^2]^2)/(-12*x + 4*E^4*x - 4*x^2 + (-12*x + 4*E^4*x - 4*x^2)*Log[3*x - E^4*x + x^2] + (-3*x + E^4*x - x^2)*Log[3*x - E^4*x + x^2]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {3+2 x-36 x^3-12 x^4+e^4 \left (-1+12 x^3\right )+\left (-36 x^3+12 e^4 x^3-12 x^4\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-9 x^3+3 e^4 x^3-3 x^4\right ) \log ^2\left (3 x-e^4 x+x^2\right )}{-12 x+4 e^4 x-4 x^2+\left (-12 x+4 e^4 x-4 x^2\right ) \log \left (3 x-e^4 x+x^2\right )+\left (-3 x+e^4 x-x^2\right ) \log ^2\left (3 x-e^4 x+x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ x^3+\frac {1}{\log \left (x \left (x-e^4+3\right )\right )+2} \]