\[ \int \frac {-4 x+8 x^2+20 x^3+\left (-3 x-5 x^2\right ) \log (\log (2))+\left (-8 x-16 x^2+(2+4 x) \log (\log (2))\right ) \log \left (\frac {1}{4} (4 x-\log (\log (2)))\right )}{4 x^3+4 x^4+\left (-x^2-x^3\right ) \log (\log (2))+\left (-4 x^2-4 x^3+\left (x+x^2\right ) \log (\log (2))\right ) \log \left (\frac {1}{4} (4 x-\log (\log (2)))\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (4 \left (x -\ln \! \left (-\frac {\ln \! \left (\ln \! \left (2\right )\right )}{4}+x \right )\right ) x^{2} \left (1+x \right )^{2}\right ) \]
command
Int[(-4*x + 8*x^2 + 20*x^3 + (-3*x - 5*x^2)*Log[Log[2]] + (-8*x - 16*x^2 + (2 + 4*x)*Log[Log[2]])*Log[(4*x - Log[Log[2]])/4])/(4*x^3 + 4*x^4 + (-x^2 - x^3)*Log[Log[2]] + (-4*x^2 - 4*x^3 + (x + x^2)*Log[Log[2]])*Log[(4*x - Log[Log[2]])/4]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-4 x+8 x^2+20 x^3+\left (-3 x-5 x^2\right ) \log (\log (2))+\left (-8 x-16 x^2+(2+4 x) \log (\log (2))\right ) \log \left (\frac {1}{4} (4 x-\log (\log (2)))\right )}{4 x^3+4 x^4+\left (-x^2-x^3\right ) \log (\log (2))+\left (-4 x^2-4 x^3+\left (x+x^2\right ) \log (\log (2))\right ) \log \left (\frac {1}{4} (4 x-\log (\log (2)))\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ 2 \log (x)+2 \log (x+1)+\log \left (x-\log \left (x-\frac {1}{4} \log (\log (2))\right )\right ) \]