\[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx \]
Optimal antiderivative \[ 2+3 \left (\frac {2}{x^{2}+x -3}+3\right )^{2}+3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}} \]
command
Int[(E^x*(84 + 132*x - 108*x^2 - 72*x^3) + E^(2 + E^E^(2 - x) + E^(2 - x))*(81 - 81*x - 54*x^2 + 51*x^3 + 18*x^4 - 9*x^5 - 3*x^6))/(E^x*(-27 + 27*x + 18*x^2 - 17*x^3 - 6*x^4 + 3*x^5 + x^6)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {36}{-x^2-x+3}+\frac {12}{\left (-x^2-x+3\right )^2}+3 e^{e^{e^{2-x}}} \]