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