Thanks to sound remarks [here](https://mathoverflow.net/questions/271936/closed-form-for-int-0t-e-x-fraci-n-alpha-xxdx?rq=1) and [here](https://mathoverflow.net/questions/289163/what-can-we-know-about-the-half-of-the-generating-series-of-bessel-function), 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](https://mathoverflow.net/questions/271936/closed-form-for-int-0t-e-x-fraci-n-alpha-xxdx?rq=1), probably hypergeometric series?



Literature about the integral of Bessel $\int_0^x I_{0,1}(u) e^{-a u}du$?
The integral is studied in [On Certain Indefinite Integrals Involving Bessel Functions](http://onlinelibrary.wiley.com/doi/10.1002/sapm1958371157/abstract) (1958).  

(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)  

The paper is behind a paywall, so I have not read it.


And then there is [Tables of some indefinite integral of Bessel functions of integer order](http://web.eah-jena.de/~rsh/forschung/stoer/besint.pdf) (2017), which examines the $i=0$ integral in great detail and gives several series expansions, see pages 87 and following.


I reproduce one such formula:


![](https://ilorentz.org/beenakker/MO/Bessel_series_5.png)


