Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Given that following two random variables $$\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$$ and $$\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$$ are independent, what would be the expectation

$$\mathbb{E} \left[ \left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2 \right],$$ where $$\mathcal{CN}(.,.)$$ is the complex normal random variable.

• This is not a homework problem, it arises when calculating the spectral efficiency of an i.i.d. Rayleigh channel with a channel estimator proposed in earlier work. – Felipe Augusto de Figueiredo Jul 31 '19 at 14:15

Using the same trick from another answer, as well as the trace trick and $$E[1/\Vert z\Vert^2]$$ from yet another answer, we find
$$\frac{\sigma_x^2 \, \sigma_y^2}{(\sigma_x^2+\sigma_y^2)^2} \, \frac{1}{M-1} + \frac{(\sigma_x^2)^2}{(\sigma_x^2+\sigma_y^2)^2} \;.$$
$$\mathbb{E} \left[ \left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2 \right] \approx \frac{\mathbb{E} \left[ \|(\textbf{x}+\textbf{y})^{H} \textbf{x} \|^2 \right]}{ \mathbb{E} \left[ \| \textbf{x} + \textbf{y} \|^4 \right]} = \frac{(M+1)(\sigma_{x}^2)^2 +\sigma_{x}^2 \sigma_{y}^2}{(M+1)(\sigma_{x}^2 + \sigma_{y}^2)^2},$$ which produces results close to the simulated expectation.