$\mathbb{E} \left[ \frac{X}{c + X} \right]$ for known constant $c > 0$? In general, describing expectations of ratios of random variables can be tricky. I have a ratio of random variables, but thankfully, it's nicely behaved due to known structure. Specifically, I have a univariate random variable $X$ whose support is non-negative reals $\mathbb{R}_{\geq 0}$, and I want to compute
$$\mathbb{E} \Bigg[ \frac{X}{c + X} \Bigg]$$
where $c > 0$ is a known constant. Is there an exact expression for this expectation, perhaps in terms of the moments (or centered moments) of $X$?
Edit: The first-order Taylor series approximation is $\frac{\mathbb{E}_X[X]}{c + \mathbb{E}_X[X]}$ but I was hoping for something better / exact if possible.

We can rewrite the expression as:
$$E{\frac {X}{X+c}}=1-cE{\frac {1}{X+c}}=1-c\int {\frac 1 {x+c}}p_X(dx)=1-cH_{p_X}(-c)$$
where $H_{p}$ denotes the Hilbert transform of the density $p$. The wikipedia page gives a few examples of distributions for which the Hilbert transform has a closed form (in particular, the case of uniform distributions), but there is not any kind of explicit expression in general.
