22.12 Problem number 1020

$$\int \frac{e^{\frac{x}{1+x}}(x^2+(-5-5x-6x^2)\log (2))+e^{\frac{2x}{1+x}}(-2-4x-2x^2-243x^4-486x^5-243x^6+(2+4x+2x^2-1620x^3-2997x^4-1134x^5+243x^6)\log (2))}{(x^2+2x^3+x^4)\log (2)+e^{\frac{x}{1+x}}(-4x-8x^2-4x^3+162x^5+324x^6+162x^7)\log (2)+e^{\frac{2x}{1+x}}(4+8x+4x^2-324x^4-648x^5-324x^6+6561x^8+13122x^9+6561x^{10})\log (2)}dx$$

Optimal antiderivative $$\frac{\frac{x}{\ln \left(2\right)}+5-x}{x{\mathrm{e}}^{-\frac{x}{1+x}}-2+81x^4}$$

command

Integrate[(E^(x/(1 + x))*(x^2 + (-5 - 5*x - 6*x^2)*Log[2]) + E^((2*x)/(1 + x))*(-2 - 4*x - 2*x^2 - 243*x^4 - 486*x^5 - 243*x^6 + (2 + 4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6)*Log[2]))/((x^2 + 2*x^3 + x^4)*Log[2] + E^(x/(1 + x))*(-4*x - 8*x^2 - 4*x^3 + 162*x^5 + 324*x^6 + 162*x^7)*Log[2] + E^((2*x)/(1 + x))*(4 + 8*x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10)*Log[2]),x]

Mathematica 13.1 output

$$\int \frac{e^{\frac{x}{1+x}}(x^2+(-5-5x-6x^2)\log (2))+e^{\frac{2x}{1+x}}(-2-4x-2x^2-243x^4-486x^5-243x^6+(2+4x+2x^2-1620x^3-2997x^4-1134x^5+243x^6)\log (2))}{(x^2+2x^3+x^4)\log (2)+e^{\frac{x}{1+x}}(-4x-8x^2-4x^3+162x^5+324x^6+162x^7)\log (2)+e^{\frac{2x}{1+x}}(4+8x+4x^2-324x^4-648x^5-324x^6+6561x^8+13122x^9+6561x^{10})\log (2)}dx$$

Mathematica 12.3 output

$$\frac{3944312x+19785288\log (2)-1524858x\log (2)+72868\log (4)-595423x\log (4)-36434\log (16)+51192x\log (16)-9104\log (128)-204768x\log (128)+\frac{3944312e^{\frac{1}{1+x}}x(243x^7(-1+\log (2))-10\log (2)-1215x^4\log (2)-81x^5(3+32\log (2))+x^3(-2+\log (4))-x(2+\log (256))-x^2(2+\log (256))-81x^6(7+\log (256)))}{(2+2x+2x^2+243x^4+567x^5+243x^6)(e^{\frac{1}{1+x}}x+e(-2+81x^4))}}{3944312(-2+81x^4)\log (2)}$$