I know how to estimate the integral \begin{gather} \int f(Ub)\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{gather} where $f:S^n(\mathbb{R})\to \mathbb{R}$ and $f(x)=\sum\limits_{i=1}^{n}\frac{1}{x_i^2}$ (which is $\infty$) and $\mu$ 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. $$\int g(UMb)\mu(U),$$ where $g:S^n(\mathbb{C})\to \mathbb{R}$ 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 [1], 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.