\[ \int \frac {e^{2 e^x} \left (6-3 e^4\right )+12 x^2-3 e^4 x^2-3 x^2 \log (2)+\left (-16 x+6 e^4 x+2 x \log (2)\right ) \log (3)+\left (6-3 e^4\right ) \log ^2(3)+e^{e^x} \left (-16 x+6 e^4 x+2 x \log (2)+e^x \left (-2 x^2+x^2 \log (2)\right )+\left (12-6 e^4\right ) \log (3)\right )}{e^{2 e^x} x^4+x^6-2 x^5 \log (3)+x^4 \log ^2(3)+e^{e^x} \left (-2 x^5+2 x^4 \log (3)\right )} \, dx \]
Optimal antiderivative \[ \frac {\frac {\left (2-\ln \left (2\right )\right ) x}{\ln \left (3\right )+{\mathrm e}^{{\mathrm e}^{x}}-x}+{\mathrm e}^{4}-2}{x^{3}} \]
command
Integrate[(E^(2*E^x)*(6 - 3*E^4) + 12*x^2 - 3*E^4*x^2 - 3*x^2*Log[2] + (-16*x + 6*E^4*x + 2*x*Log[2])*Log[3] + (6 - 3*E^4)*Log[3]^2 + E^E^x*(-16*x + 6*E^4*x + 2*x*Log[2] + E^x*(-2*x^2 + x^2*Log[2]) + (12 - 6*E^4)*Log[3]))/(E^(2*E^x)*x^4 + x^6 - 2*x^5*Log[3] + x^4*Log[3]^2 + E^E^x*(-2*x^5 + 2*x^4*Log[3])),x]
Mathematica 13.1 output
\[ \int \frac {e^{2 e^x} \left (6-3 e^4\right )+12 x^2-3 e^4 x^2-3 x^2 \log (2)+\left (-16 x+6 e^4 x+2 x \log (2)\right ) \log (3)+\left (6-3 e^4\right ) \log ^2(3)+e^{e^x} \left (-16 x+6 e^4 x+2 x \log (2)+e^x \left (-2 x^2+x^2 \log (2)\right )+\left (12-6 e^4\right ) \log (3)\right )}{e^{2 e^x} x^4+x^6-2 x^5 \log (3)+x^4 \log ^2(3)+e^{e^x} \left (-2 x^5+2 x^4 \log (3)\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {-2+e^4-\frac {x \left (2-\log (2)+e^x (x (-2+\log (2))-\log (2) \log (3)+\log (9))\right )}{\left (-1+e^x (x-\log (3))\right ) \left (e^{e^x}-x+\log (3)\right )}}{x^3} \]