$\mathbb{E} \left[ \frac{X}{c + X} \right]$ 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.
 A: Breaking down
$$
\frac{X}{c+X} = 1 - \frac{1}{1+X/c},
$$
and expanding in Taylor series, its expectation is
$$
E \left[ \frac{X}{c+X} \right] = 1 - E \left[ 1 - \frac{X}{c} + \frac{X^2}{c^2} - \frac{X^3}{c^3} + \cdots \right] \\
= E \left[ \frac{X}{c} - \frac{X^2}{c^2} + \frac{X^3}{c^3} - \cdots \right] \\
= \sum_{k=1}^\infty \frac{(-1)^{k-1}}{c^k} E[X^k]
$$
Now, that Taylor series only converges for $-c < X < c$, and so if the support of $X$ lies outside of this range, this expansion is invalid.
A: 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.
A: A general approximation without knowing more about the distribution of $X\ge 0$ seems difficult, you might be better of trying to approximate the distribution with some known distribution family, and then evaluating the expectation for that family.   I will give a few examples below, but without giving details pf the integration.
Half-Cauchy distribution
$$
\frac{2 c \ln \! \left(c \right)+\pi}{\pi  \left(c^{2}+1\right)}
$$
Uniform distribution on $[0,a]$
$$
   -\frac{c \ln \! \left(a +c \right)-c \ln \! \left(c \right)-a}{a}
$$
Exponential distribution with mean 1
$$
1+\frac{\left(-2 \,\mathrm{Ei}_{1}\! \left(c \right)-\ln \! \left(c \right)-\ln \! \left(\frac{1}{c}\right)\right) {\mathrm e}^{c} c}{2}
$$ where $\mathrm{Ei}$ is the Exponential integral.
Gamma distribution with scale 1 and shape $k>0$
$$
c^{k} {\mathrm e}^{c} k \Gamma \! \left(-k , c\right)
$$
and such results can be extended further, for instance with mixture modeling.
