\[ \int \frac {-4096 x-64 x^3+\left (-256 x+60 x^3+x^5\right ) \log \left (-4+x^2\right )+\left (4096 x+\left (256 x-64 x^3\right ) \log \left (-4+x^2\right )\right ) \log \left (\log \left (-4+x^2\right )\right )}{\left (-4096+1024 x^2\right ) \log \left (-4+x^2\right )} \, dx \]
Optimal antiderivative \[ \left (1+\frac {x^{2}}{64}-\ln \! \left (\ln \! \left (x^{2}-4\right )\right )\right )^{2} \]
command
Int[(-4096*x - 64*x^3 + (-256*x + 60*x^3 + x^5)*Log[-4 + x^2] + (4096*x + (256*x - 64*x^3)*Log[-4 + x^2])*Log[Log[-4 + x^2]])/((-4096 + 1024*x^2)*Log[-4 + x^2]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-4096 x-64 x^3+\left (-256 x+60 x^3+x^5\right ) \log \left (-4+x^2\right )+\left (4096 x+\left (256 x-64 x^3\right ) \log \left (-4+x^2\right )\right ) \log \left (\log \left (-4+x^2\right )\right )}{\left (-4096+1024 x^2\right ) \log \left (-4+x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {\left (x^2-64 \log \left (\log \left (x^2-4\right )\right )+64\right )^2}{4096} \]