\[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx \]
Optimal antiderivative \[ \frac {3}{\left (\frac {\ln \left (2\right )^{2}}{x^{2}}+1-x^{4}+\frac {1}{x}\right )^{2}} \]
command
Int[(6*x^4 + 24*x^9 + 12*x^3*Log[2]^2)/(x^3 + 3*x^4 + 3*x^5 + x^6 - 3*x^8 - 6*x^9 - 3*x^10 + 3*x^13 + 3*x^14 - x^18 + (3*x^2 + 6*x^3 + 3*x^4 - 6*x^7 - 6*x^8 + 3*x^12)*Log[2]^2 + (3*x + 3*x^2 - 3*x^6)*Log[2]^4 + Log[2]^6),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {6 x^4+24 x^9+12 x^3 \log ^2(2)}{x^3+3 x^4+3 x^5+x^6-3 x^8-6 x^9-3 x^{10}+3 x^{13}+3 x^{14}-x^{18}+\left (3 x^2+6 x^3+3 x^4-6 x^7-6 x^8+3 x^{12}\right ) \log ^2(2)+\left (3 x+3 x^2-3 x^6\right ) \log ^4(2)+\log ^6(2)} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {3 x^4}{\left (-x^6+x^2+x+\log ^2(2)\right )^2} \]