Currently, I'm working on a problem pertaining to certain integrals involving the modified Bessel function of the first kind. On p. 59 of this book by Rosenheinrich, it is stated that

$$\int e^{-x} I_{0}(x) dx = x e^{-x}[I_{0}(x) + I_{1}(x)] .$$

Moreover, Wolfram obtains: $$\int I_{0}(2 \sqrt{x}) = \sqrt{x} I_{1}(2 \sqrt{x}) + c. $$

Definite integrals are also of interest to me. For instance , we have: $$\int_{0}^{\infty} e^{-x} I_{0}(2 \sqrt{x}) dx = e. $$

Now, I'm looking for results on the following integral:

$$\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx \qquad \qquad (*) $$

The definite version for $x$ ranging over the nonnegative reals amounts to about $0,761$.

Question: have any results been obtained on the integral stated in $(*)$ - with regards to both the indefinite and definite version?