How do I calculate the expectation of the rational function, in the sense of the Haar measure?

Question

I want to know the analytical solution of $\mathbb{E}_{\psi}\frac{\langle \psi |A|\psi\rangle}{\langle \psi |A^2|\psi\rangle}$. I see similar questions before approximate average, but it does not provide a very specific calculation method. So I want to know if there is any method to calculate this expectation, and it would be best if some simple examples being given, such as when A is a density operator. If not, is there anything similar in classical probability theory? Please advise.