22.9 Problem number 414

$$\int \frac{e^{2e^x}(6-3e^4)+12x^2-3e^4{x}^2-3x^2\log (2)+(-16x+6e^{4}x+2x\log (2))\log (3)+(6-3e^4){\log }^2(3)+e^{e^x}(-16x+6e^{4}x+2x\log (2)+e^x(-2x^2+x^2\log (2))+(12-6e^4)\log (3))}{e^{2e^x}{x}^4+x^6-2x^5\log (3)+x^4{\log }^2(3)+e^{e^x}(-2x^5+2x^4\log (3))}dx$$

Optimal antiderivative $$\frac{\frac{(2-\ln \left(2\right))x}{\ln \left(3\right)+{\mathrm{e}}^{{\mathrm{e}}^x}-x}+{\mathrm{e}}^4-2}{x^3}$$

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

$$\int \frac{e^{2e^x}(6-3e^4)+12x^2-3e^4{x}^2-3x^2\log (2)+(-16x+6e^{4}x+2x\log (2))\log (3)+(6-3e^4){\log }^2(3)+e^{e^x}(-16x+6e^{4}x+2x\log (2)+e^x(-2x^2+x^2\log (2))+(12-6e^4)\log (3))}{e^{2e^x}{x}^4+x^6-2x^5\log (3)+x^4{\log }^2(3)+e^{e^x}(-2x^5+2x^4\log (3))}dx$$

Mathematica 12.3 output

$$\frac{-2+e^4-\frac{x(2-\log (2)+e^x(x(-2+\log (2))-\log (2)\log (3)+\log (9)))}{(-1+e^x(x-\log (3)))(e^{e^x}-x+\log (3))}}{x^3}$$