\[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\ln \! \left (4 x +\frac {3}{\ln \! \left (\ln \! \left (x +\frac {9}{x}+1\right )\right )}-5\right )^{2}\right ) \]
command
Int[(54 - 6*x^2 + (72*x + 8*x^2 + 8*x^3)*Log[(9 + x + x^2)/x]*Log[Log[(9 + x + x^2)/x]]^2)/(((27*x + 3*x^2 + 3*x^3)*Log[(9 + x + x^2)/x]*Log[Log[(9 + x + x^2)/x]] + (-45*x + 31*x^2 - x^3 + 4*x^4)*Log[(9 + x + x^2)/x]*Log[Log[(9 + x + x^2)/x]]^2)*Log[(3 + (-5 + 4*x)*Log[Log[(9 + x + x^2)/x]])/Log[Log[(9 + x + x^2)/x]]]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ 2 \log \left (\log \left (4 x+\frac {3}{\log \left (\log \left (x+\frac {9}{x}+1\right )\right )}-5\right )\right ) \]