\[ \int \frac {e^{\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )} \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx \]
Optimal antiderivative \[ 1+3 \,{\mathrm e}^{-{\left (-\frac {3}{2}+2 x +\ln \left (-x^{2}+4 x \right )\right )}^{2}} \]
command
Int[(E^((-9 + 24*x - 16*x^2 - (-12 + 16*x)*Log[4*x - x^2] - 4*Log[4*x - x^2]^2)/4)*(-36 - 6*x + 90*x^2 - 24*x^3 + (24 + 36*x - 12*x^2)*Log[4*x - x^2]))/(-4*x + x^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ 3 \exp \left (-\frac {1}{4} (-4 x-2 \log ((4-x) x)+3)^2\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {\exp \left (\frac {1}{4} \left (-9+24 x-16 x^2-(-12+16 x) \log \left (4 x-x^2\right )-4 \log ^2\left (4 x-x^2\right )\right )\right ) \left (-36-6 x+90 x^2-24 x^3+\left (24+36 x-12 x^2\right ) \log \left (4 x-x^2\right )\right )}{-4 x+x^2} \, dx \]________________________________________________________________________________________