Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is: $$\int_{O(N)}U_{11}^aU_{22}^bU_{33}^cdU=?$$

This is quite a tricky question:

(1) The first thought would go to probability, because with $N\to\infty$ the variables $U_{11},U_{22},U_{33}$ become Gaussian and independent; however, there doesn't seem to be any good analytic method for computing the correlations at $N$ fixed.

(2) The second thought would go to combinatorics, and to the Weingarten function. But that doesn't work either: we spent some time with Collins and Schlenker on this question, and just got a kind of very long (and especially unusable!) formula.

So, a new point of view on all this would be probably needed.