\[ \int \frac {e^{-e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}}} \left (64+32 x+36 x^2+8 x^3+4 x^4+e^{\frac {80+e^x+36 x+24 x^2+4 x^3}{16+4 x+4 x^2}} \left (-64 x-32 x^2-36 x^3-8 x^4-4 x^5+e^x \left (-3 x+x^2-x^3\right )\right )\right )}{16+8 x+9 x^2+2 x^3+x^4} \, dx \]
Optimal antiderivative \[ 3+4 x \,{\mathrm e}^{-{\mathrm e}^{x +\frac {{\mathrm e}^{x}}{4 x^{2}+4 x +16}+5}} \]
command
Int[(64 + 32*x + 36*x^2 + 8*x^3 + 4*x^4 + E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 + 4*x + 4*x^2))*(-64*x - 32*x^2 - 36*x^3 - 8*x^4 - 4*x^5 + E^x*(-3*x + x^2 - x^3)))/(E^E^((80 + E^x + 36*x + 24*x^2 + 4*x^3)/(16 + 4*x + 4*x^2))*(16 + 8*x + 9*x^2 + 2*x^3 + x^4)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {4 \left (4 x^5+8 x^4+36 x^3+32 x^2+e^x \left (x^3-x^2+3 x\right )+64 x\right ) \exp \left (-e^{\frac {4 x^3+24 x^2+36 x+e^x+80}{4 \left (x^2+x+4\right )}}\right )}{\left (x^4+2 x^3+9 x^2+8 x+16\right ) \left (\frac {12 x^2+48 x+e^x+36}{x^2+x+4}-\frac {(2 x+1) \left (4 x^3+24 x^2+36 x+e^x+80\right )}{\left (x^2+x+4\right )^2}\right )} \]