\[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx \]
Optimal antiderivative \[ 2 \ln \! \left ({\ln \! \left (x +\frac {9}{x \left (\left (x^{2}+1\right )^{2}+5\right )}\right )}^{2}+\frac {1}{4}\right ) \]
command
Int[((-864 - 288*x^2 - 336*x^4 + 256*x^6 + 64*x^8 + 16*x^10)*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)])/(54*x + 54*x^3 + 33*x^5 + 16*x^7 + 4*x^9 + x^11 + (216*x + 216*x^3 + 132*x^5 + 64*x^7 + 16*x^9 + 4*x^11)*Log[(9 + 6*x^2 + 2*x^4 + x^6)/(6*x + 2*x^3 + x^5)]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (-864-288 x^2-336 x^4+256 x^6+64 x^8+16 x^{10}\right ) \log \left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )}{54 x+54 x^3+33 x^5+16 x^7+4 x^9+x^{11}+\left (216 x+216 x^3+132 x^5+64 x^7+16 x^9+4 x^{11}\right ) \log ^2\left (\frac {9+6 x^2+2 x^4+x^6}{6 x+2 x^3+x^5}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ 2 \log \left (4 \log ^2\left (\frac {x^6+2 x^4+6 x^2+9}{x^5+2 x^3+6 x}\right )+1\right ) \]