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