\[ \int \frac {\left (4 x^6+8 x^3 \log (4)\right ) \log \left (\frac {121 x^4-110 x^5+25 x^6+\left (110 x^2-50 x^3\right ) \log (4)+25 \log ^2(4)}{4 x^4-4 x^5+x^6+\left (4 x^2-2 x^3\right ) \log (4)+\log ^2(4)}\right )+\left (44 x^5-42 x^6+10 x^7+\left (42 x^3-20 x^4\right ) \log (4)+10 x \log ^2(4)\right ) \log ^2\left (\frac {121 x^4-110 x^5+25 x^6+\left (110 x^2-50 x^3\right ) \log (4)+25 \log ^2(4)}{4 x^4-4 x^5+x^6+\left (4 x^2-2 x^3\right ) \log (4)+\log ^2(4)}\right )}{22 x^4-21 x^5+5 x^6+\left (21 x^2-10 x^3\right ) \log (4)+5 \log ^2(4)} \, dx \]
Optimal antiderivative \[ {\ln \! \left (\left (-5+\frac {x}{x^{2}-2 x -\frac {2 \ln \left (2\right )}{x}}\right )^{2}\right )}^{2} x^{2} \]
command
Int[((4*x^6 + 8*x^3*Log[4])*Log[(121*x^4 - 110*x^5 + 25*x^6 + (110*x^2 - 50*x^3)*Log[4] + 25*Log[4]^2)/(4*x^4 - 4*x^5 + x^6 + (4*x^2 - 2*x^3)*Log[4] + Log[4]^2)] + (44*x^5 - 42*x^6 + 10*x^7 + (42*x^3 - 20*x^4)*Log[4] + 10*x*Log[4]^2)*Log[(121*x^4 - 110*x^5 + 25*x^6 + (110*x^2 - 50*x^3)*Log[4] + 25*Log[4]^2)/(4*x^4 - 4*x^5 + x^6 + (4*x^2 - 2*x^3)*Log[4] + Log[4]^2)]^2)/(22*x^4 - 21*x^5 + 5*x^6 + (21*x^2 - 10*x^3)*Log[4] + 5*Log[4]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ x^2 \log ^2\left (\frac {\left (-5 x^3+11 x^2+5 \log (4)\right )^2}{\left (-x^3+2 x^2+\log (4)\right )^2}\right ) \]