\[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx \]
Optimal antiderivative \[ \left (x -\frac {x}{-\frac {\ln \left (x \right ) x^{2}}{\ln \left (3\right )}+2}\right )^{2} \]
command
Integrate[(2*x^3*Log[3]^2 - 4*x*Log[3]^3 + (-2*x^5*Log[3] + 14*x^3*Log[3]^2)*Log[x] - 12*x^5*Log[3]*Log[x]^2 + 2*x^7*Log[x]^3)/(-8*Log[3]^3 + 12*x^2*Log[3]^2*Log[x] - 6*x^4*Log[3]*Log[x]^2 + x^6*Log[x]^3),x]
Mathematica 13.1 output
\[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx \]
Mathematica 12.3 output
\[ \frac {x^2 \left (8 \log ^2(3) \log ^3(9)-x^6 \left (\log ^2(3)+\log (3) \log (9)-\log ^2(9)\right )+4 x^2 \log ^2(9) \left (\log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )+x^4 \left (\log ^3(9)+\log ^2(3) \log (81)\right )-\left (x^8 \log (9)+32 x^2 \log ^2(3) \log ^2(9)+6 x^6 \left (2 \log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )+8 x^4 \log (9) \left (4 \log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )\right ) \log (x)+x^4 \left (x^2+\log (81)\right )^3 \log ^2(x)\right )}{\left (x^2+\log (81)\right )^3 \left (\log (9)-x^2 \log (x)\right )^2} \]