Let $\sigma_{n, m}$ denote the uniform measure over matrices $U \in \mathbb{R}^{n \times m}$ satisfying $U^T U = I_m$. Let $1_k \in \mathbb{R}^k$ denote the vector with all entries equal to $1$. I am trying to calculate, for $m < n$, the following quantity $$ \mathcal{I}(n, m) = \int \frac{1}{1 - \tfrac{1}{n}\langle U^T 1_n, U^T 1_n \rangle} \, d \sigma_{n,m}(U) = \int\frac{1}{1 - \langle U^T v, U^T v \rangle } d\sigma_{n,m}(U), $$ where the equality holds for any $v \in \mathbb{R}^n$ with unit Euclidean norm.

Is there a convenient way to calculate this Haar integral?

Let $\sigma_{n, m}$ denote the uniform measure over matrices $U \in \mathbb{R}^{n \times m}$ satisfying $U^T U = I_m$. Let $1_k \in \mathbb{R}^k$ denote the vector with all entries equal to $1$. I am trying to calculate, for $m < n$, the following quantity $$ \mathcal{I}(n, m) = \int \frac{1}{1 - \tfrac{1}{n}\langle U^T 1_n, U^T 1_n \rangle} \, d \sigma_{n,m}(U) = \int\frac{1}{1 - \langle U^T v, U^T v \rangle } d\sigma_{n,m}(U), $$ where the final equality holds for any $v \in \mathbb{R}^n$ with unit Euclidean norm, simply using that $\sigma_{n, m}$ is unitarily invariant.

Is there a convenient way to calculate this Haar integral?