\[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx \]
Optimal antiderivative \[ 3-\left ({\mathrm e}^{\frac {-2+x}{\left (x -{\mathrm e}\right ) \left (1+2 x \right )}}+1\right )^{2}-x \]
command
Int[(-x^2 - 4*x^3 - 4*x^4 + E^2*(-1 - 4*x - 4*x^2) + E^((2 - x)/(-x - 2*x^2 + E*(1 + 2*x)))*(-4 + 10*E - 16*x + 4*x^2) + E^((2*(2 - x))/(-x - 2*x^2 + E*(1 + 2*x)))*(-4 + 10*E - 16*x + 4*x^2) + E*(2*x + 8*x^2 + 8*x^3))/(x^2 + 4*x^3 + 4*x^4 + E^2*(1 + 4*x + 4*x^2) + E*(-2*x - 8*x^2 - 8*x^3)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-x^2-4 x^3-4 x^4+e^2 \left (-1-4 x-4 x^2\right )+e^{\frac {2-x}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e^{\frac {2 (2-x)}{-x-2 x^2+e (1+2 x)}} \left (-4+10 e-16 x+4 x^2\right )+e \left (2 x+8 x^2+8 x^3\right )}{x^2+4 x^3+4 x^4+e^2 \left (1+4 x+4 x^2\right )+e \left (-2 x-8 x^2-8 x^3\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -x-2 e^{\frac {2-x}{(e-x) (2 x+1)}}-e^{\frac {2 (2-x)}{(e-x) (2 x+1)}}-\frac {4 e^4}{(1+2 e)^2 (e-x)}-\frac {4 e^3}{(1+2 e)^2 (e-x)}-\frac {e^2}{(1+2 e)^2 (e-x)}+\frac {e^2}{e-x}-\frac {4 e^2 (3+2 e) \log (e-x)}{(1+2 e)^3}-\frac {16 e^3 (1+e) \log (e-x)}{(1+2 e)^3}-\frac {2 e \log (e-x)}{(1+2 e)^3}+2 e \log (e-x)-\frac {(1+6 e) \log (2 x+1)}{(1+2 e)^3}+\frac {(1+4 e) \log (2 x+1)}{(1+2 e)^3}+\frac {2 e \log (2 x+1)}{(1+2 e)^3} \]