I need to find the expected value of the following type of an expression: $$\mathbb{E}[\frac{1-\alpha}{1-\alpha-\frac{X}{\beta}}]$$ where $\alpha$ and $\beta$ are constants and $X$ is a random variable. I know that $X$ is distributed as a doubly truncated normal with mean $\xi$ and standard deviation $\sigma$, the lower and upper truncation points being $A$ and $B$ respectively. For the problem to be well defined, assume $A<B<\beta(1-\alpha)$.
I know the pdf of the truncated normal with the above parameters is:
I also know the expected value of my variable $X$ is:
The latter is clearly not very helpful here due to Jensen's inequality.
So far I simulated the sample equivalent for particular parameters and that works fine for me (I can bootstrap a confidence interval afterwards for stability). I am trying to work out whether I can find a nice expression like the one for $\mathbb{E}[X]$ above though - any ideas?