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