\[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx \]
Optimal antiderivative \[ \ln \! \left (\frac {x}{x +\frac {16}{x^{4}}}+x \right )+\ln \! \left (\ln \! \left (x \right )\right ) \]
command
Int[(256 + 16*x^4 + 32*x^5 + x^9 + x^10 + (256 + 80*x^4 + 32*x^5 + x^10)*Log[x])/((256*x + 16*x^5 + 32*x^6 + x^10 + x^11)*Log[x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\log \left (x^5+16\right )+\log \left (x^4-x^3+2 x^2-4 x+8\right )+\log (x)+\log (x+2)+\log (\log (x)) \]