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