Suppose $X_1, X_2, ..., X_N$ are i.i.d with Gaussian distribution with unit mean and variance $\sigma^2$, can we find the following expectation ?
\begin{equation} \mathbb{E} \left[\frac{1}{1+X_1^2+X_2^2+...+X_N^2}\right] \end{equation}
Actually I wanted to calculate following expectation, then I realized I need that :
\begin{align} &\mathbb{E} \left[\frac{X_1^2+X_2^2+...+X_N^2}{1+X_1^2+X_2^2+...+X_N^2}\right]\\ &\qquad=\mathbb{E} \left[\frac{1+X_1^2+X_2^2+...+X_N^2}{1+X_1^2+X_2^2+...+X_N^2}\right]-\mathbb{E} \left[\frac{1}{1+X_1^2+X_2^2+...+X_N^2}\right]\\ &\qquad=1-\mathbb{E} \left[\frac{1}{1+X_1^2+X_2^2+...+X_N^2}\right] \end{align}