\[ \int \frac {-4+13 x-11 x^2+2 x^3+\left (4-9 x+2 x^2\right ) \log (4)+\left (-2-3 x+x^2+x^3+\left (-2-2 x+2 x^2\right ) \log (4)\right ) \log \left (1+x-x^2\right )}{\left (-4+x+8 x^2-6 x^3+x^4+\left (4+3 x-5 x^2+x^3\right ) \log (4)\right ) \log \left (1+x-x^2\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\frac {\left (-x +4\right )^{2} \ln \! \left (-x^{2}+x +1\right )}{x +2 \ln \! \left (2\right )-1}\right ) \]
command
Int[(-4 + 13*x - 11*x^2 + 2*x^3 + (4 - 9*x + 2*x^2)*Log[4] + (-2 - 3*x + x^2 + x^3 + (-2 - 2*x + 2*x^2)*Log[4])*Log[1 + x - x^2])/((-4 + x + 8*x^2 - 6*x^3 + x^4 + (4 + 3*x - 5*x^2 + x^3)*Log[4])*Log[1 + x - x^2]),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-4+13 x-11 x^2+2 x^3+\left (4-9 x+2 x^2\right ) \log (4)+\left (-2-3 x+x^2+x^3+\left (-2-2 x+2 x^2\right ) \log (4)\right ) \log \left (1+x-x^2\right )}{\left (-4+x+8 x^2-6 x^3+x^4+\left (4+3 x-5 x^2+x^3\right ) \log (4)\right ) \log \left (1+x-x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log \left (\log \left (-x^2+x+1\right )\right )+\frac {(6+\log (16)) \log (4-x)}{3+\log (4)}-\log (-x+1-\log (4)) \]