\[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx \]
Optimal antiderivative \[ 3 \,{\mathrm e}^{-x^{2} \left ({\mathrm e}^{x}-{\mathrm e}^{2 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}+2 x}\right )^{4}} \]
command
Int[E^(-(E^(4*x)*x^2) + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2 - E^(8*E^(4 - E^x^2) + 8*x)*x^2)*(E^(4*x)*(-6*x - 12*x^2) + E^(8*E^(4 - E^x^2) + 8*x)*(-6*x - 24*x^2 + 48*E^(4 - E^x^2 + x^2)*x^3) + E^(4*E^(4 - E^x^2) + 4*x)*(144*E^(4 - E^x^2 + 2*x + x^2)*x^3 + E^(2*x)*(-36*x - 108*x^2)) + E^(2*E^(4 - E^x^2) + 2*x)*(-48*E^(4 - E^x^2 + 3*x + x^2)*x^3 + E^(3*x)*(24*x + 60*x^2)) + E^(6*E^(4 - E^x^2) + 6*x)*(-144*E^(4 - E^x^2 + x + x^2)*x^3 + E^x*(24*x + 84*x^2))),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ 3 \exp \left (-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2\right ) \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \exp \left (-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2\right ) \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx \]________________________________________________________________________________________