Expectation of a function of a normal random variable

Question

Suppose $X\sim\mathcal{N}(0,1)$. I would like to find $\mathbb{E}[\frac{1}{\alpha+\beta X}|A<X<B]$ where $A, \alpha, \beta>0$. How should I go about it?

Finally, if the answer is that there is no closed-form solution, how can one show this?

Answer 1

It has no closed-form solution, but the expectation is the solution to the following definite integral: $\int_A^B \frac{e^{-0.5x^2}}{(\alpha+\beta x)\sqrt{2\pi}} dx$