\[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx \]
Optimal antiderivative \[ x +{\mathrm e}+8-\frac {4}{5 \,{\mathrm e}^{\left (-x \,{\mathrm e}^{3}+x \right )^{2}+6}+5 x} \]
command
Integrate[(4 + 5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 5*x^2 + E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*(18*x - 16*E^3*x + 8*E^6*x))/(5*E^(12 + 2*x^2 - 4*E^3*x^2 + 2*E^6*x^2) + 10*E^(6 + x^2 - 2*E^3*x^2 + E^6*x^2)*x + 5*x^2),x]
Mathematica 13.1 output
\[ -\frac {\left (9-8 e^3+4 e^6\right ) x \left (-3+2 x^2-20 e^9 x^2+2 e^{12} x^2+e^3 \left (8-20 x^2\right )+e^6 \left (-3+36 x^2\right )\right )}{10 \left (-1+e^3\right )^2 \left (1-2 \left (-1+e^3\right )^2 x^2\right )^2}+\frac {1}{5} \left (5 x-\frac {\left (9-8 e^3+13 e^6-8 e^9+4 e^{12}\right ) x}{\left (-1+e^3\right )^2 \left (1-2 \left (-1+e^3\right )^2 x^2\right )^2}-\frac {3 \left (9-8 e^3+13 e^6-8 e^9+4 e^{12}\right ) x}{2 \left (-1+e^3\right )^2 \left (-1+2 \left (-1+e^3\right )^2 x^2\right )}+\frac {2 e^{-2 e^3 x^2} \left (9 e^{6+\left (1+e^3\right )^2 x^2} x-8 e^{9+\left (1+e^3\right )^2 x^2} x+4 e^{12+\left (1+e^3\right )^2 x^2} x+e^{4 e^3 x^2} \left (2+5 x^2\right )\right )}{\left (e^{6+\left (1+e^6\right ) x^2}+e^{2 e^3 x^2} x\right ) \left (-1+2 \left (-1+e^3\right )^2 x^2\right )}\right ) \]
Mathematica 12.3 output
\[ \int \frac {4+5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+5 x^2+e^{6+x^2-2 e^3 x^2+e^6 x^2} \left (18 x-16 e^3 x+8 e^6 x\right )}{5 e^{12+2 x^2-4 e^3 x^2+2 e^6 x^2}+10 e^{6+x^2-2 e^3 x^2+e^6 x^2} x+5 x^2} \, dx \]________________________________________________________________________________________