Thanks to sound remarks [here][1] and [here][2], and looking again at these equations, I noticed that my puzzle boils down to these 2 special functions:
$$\int_0^x I_i(u) e^{-a u}du, \quad i=0,1$$
where $I_n(u)$ is the modified Bessel of the first kind.

Are these integral echoing something known in the literature? What can we know for $x<+\infty$ and $\alpha >1$? Especially for finite $x$, I know that it can be expanded in powers of $\frac{a^{-2}-1}{2}$ cf. [here][1], probably hypergeometric series?

  [1]: https://mathoverflow.net/questions/271936/closed-form-for-int-0t-e-x-fraci-n-alpha-xxdx?rq=1
  [2]: https://mathoverflow.net/questions/289163/what-can-we-know-about-the-half-of-the-generating-series-of-bessel-function