\[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx \]
Optimal antiderivative \[ \ln \! \left (30 \left (\frac {2}{2 \ln \! \left (2\right )-x}+x^{2}\right ) x \right ) \]
command
Int[(3*x^4 + (2 - 6*x^3)*Log[4] + 3*x^2*Log[4]^2)/(-2*x^2 + x^5 + (2*x - 2*x^4)*Log[4] + x^3*Log[4]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log \left (-x^3+x^2 \log (4)+2\right )+\log (x)-\log (x-\log (4)) \]