\[ \int \frac {4+x-16 x^5-8 x^6-x^7+\left (1-8 x^5-2 x^6\right ) \log (3)-x^5 \log ^2(3)+(-16-5 x-4 \log (3)) \log (x)}{-16 x^6-8 x^7-x^8+\left (-8 x^6-2 x^7\right ) \log (3)-x^6 \log ^2(3)+\left (4 x+x^2+x \log (3)\right ) \log (x)} \, dx \]
Optimal antiderivative \[ 2+\ln \left (x -\frac {\ln \left (x \right )}{\left (\ln \left (3\right )+4+x \right ) x^{4}}\right ) \]
command
Int[(4 + x - 16*x^5 - 8*x^6 - x^7 + (1 - 8*x^5 - 2*x^6)*Log[3] - x^5*Log[3]^2 + (-16 - 5*x - 4*Log[3])*Log[x])/(-16*x^6 - 8*x^7 - x^8 + (-8*x^6 - 2*x^7)*Log[3] - x^6*Log[3]^2 + (4*x + x^2 + x*Log[3])*Log[x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \log \left (x^6+x^5 (4+\log (3))-\log (x)\right )-\frac {(16+\log (81)) \log (x)}{4+\log (3)}-\log (x+4+\log (3)) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {4+x-16 x^5-8 x^6-x^7+\left (1-8 x^5-2 x^6\right ) \log (3)-x^5 \log ^2(3)+(-16-5 x-4 \log (3)) \log (x)}{-16 x^6-8 x^7-x^8+\left (-8 x^6-2 x^7\right ) \log (3)-x^6 \log ^2(3)+\left (4 x+x^2+x \log (3)\right ) \log (x)} \, dx \]________________________________________________________________________________________