Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$it can also be expressed in terms of hypergeometric function.where $\alpha$ is a positive integer.

I have solved a problem and arrived to the following function, $$F_2(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} e^{- t} dt$$

I would like to re-write it in a simplified form, does anyone know if $F_{2}(x)$ can be further simplified?