I know how to estimate the integral* (see the update) \begin{gather} \int f(Ub)d\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2] \end{gather} where $f:S^{n-1}(\mathbb{R})\to \mathbb{R}$ and $f(x)=\sum\limits_{i=1}^{n}\frac{1}{x_i^2}$ and $d\mu$ is the Haar measure over orthogonal group $O(n)$. My trouble is the following:

Suppose we multiply $b$ with a matrix of complex values (in my case DFT matrix) $M$, i.e. $$\int g(UMb)d\mu(U),$$ where $g:S^{n-1}(\mathbb{C})\to \mathbb{R}$ and $g(x)=\sum\limits_{i=1}^{n}\frac{1}{|x_i|^2}$ where $|.|$ is the absolute value. It is intuitively obvious for me that this integral should be equal to [2], specially considering that if $M$ was orthogonal this would have been true (due to properties of Haar measure), yet here $M$ is unitary.

Since my calculus knowledge is not that amazing --as you see-- I would appreciate some guidance to prove this. I imagine the title is not the best but that is what I could came up with.

** UPDATE**: This integral (as mentioned by @CarloBeenakker) is infinity (and I was aware of that). The integral I was referring (and mistakenly didn't mention) to, and have estimation of it is $\int h(Ub)\mu(U)$ where $h(x)=\sum_{i=1}^{m} \frac{1}{x_i^2 + \epsilon}$ for some positive $\epsilon$.