\[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx \]
Optimal antiderivative \[ \left (\ln \left (\frac {\left (x +\ln \left (x \right )\right )^{2}}{\left (-\frac {1}{x^{2}}+\left (3+x \right )^{2}\right )^{2}}\right )+5\right ) x \]
command
Int[(-2 - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + (-9 + 45*x^2 + 18*x^3 + x^4)*Log[x] + (-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x])*Log[(x^6 + 2*x^5*Log[x] + x^4*Log[x]^2)/(1 - 18*x^2 - 12*x^3 + 79*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8)])/(-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ x \log \left (\frac {x^4 (x+\log (x))^2}{\left (-x^4-6 x^3-9 x^2+1\right )^2}\right )+5 x+\left (3-\sqrt {5}\right ) \log \left (2 x-\sqrt {5}+3\right )+\frac {1}{12} \left (9-5 \sqrt {5}\right ) \log \left (2 x-\sqrt {5}+3\right )-\frac {1}{12} \left (45-17 \sqrt {5}\right ) \log \left (2 x-\sqrt {5}+3\right )-\frac {1}{12} \left (45+17 \sqrt {5}\right ) \log \left (2 x+\sqrt {5}+3\right )+\frac {1}{12} \left (9+5 \sqrt {5}\right ) \log \left (2 x+\sqrt {5}+3\right )+\left (3+\sqrt {5}\right ) \log \left (2 x+\sqrt {5}+3\right )+\frac {1}{12} \left (9-\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+3\right )+\left (3-\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+3\right )-\frac {1}{12} \left (45-13 \sqrt {13}\right ) \log \left (2 x-\sqrt {13}+3\right )-\frac {1}{12} \left (45+13 \sqrt {13}\right ) \log \left (2 x+\sqrt {13}+3\right )+\frac {1}{12} \left (9+\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+3\right )+\left (3+\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+3\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx \]________________________________________________________________________________________