  We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$,
depending on a matrix $a\in M_{p\times q}(\mathbb N)$, whose exact computation
is an open problem. Our results are as follows: (1) an extension of the
"elementary expansion" formula from the case $a\in M_{2\times q}(2\mathbb N)$
to the general case $a\in M_{p\times q}(\mathbb N)$, (2) the construction of
the "best algebraic normalization" of $I(a)$, in the case $a\in M_{2\times
q}(\mathbb N)$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in
M_{3\times 3}(\mathbb N)$, (4) a modelling result in the case $a\in M_{1\times
2}(\mathbb N)$, in relation with the Euler-Rodrigues formula. Most proofs use
various combinatorial techniques.
