Is there any way to evaluate this integral analytically: $$I(\Omega) = \int_{\Omega} \left(\frac{y^Tx}{\|x\|\|y\|}\right)^N dx$$ where $x \in \Omega$, $y\in P$, $\Omega\subseteq P$, the probability simplex. The $\Omega$ I'm interested in are sets such that given some $x^0$ and a set of indices $S$, $\Omega(x^0,S) =\{x:x_i=x^0_i \: \forall i\in S\}$.
My attempt so far: Normalization is a bijective transformation between the simplex and the sphere, and while $u=T(x)=x/\|x\|$ doesn't have a full rank Jacobian, the local inverse $x=u/(e^Tu)$ does, where $e$ is the vector of ones. Therefore$$I(\Omega) = \int_{T(\Omega)} \left(\frac{(y^Tu)^N}{e^Tu}-\frac{(y^Tu)^N}{(e^Tu)^2}\right) du$$ for $x\in T(\Omega)$, $y\in T(P)$, $T(\Omega)\subseteq T(P) \subset S^{d-1}$. The later seems like it might hold some advantages over the former: the square roots have disappeared and integrating over the whole simplex (a region I'm also interested in) is now integrating over the all-postive section of the sphere, which avoids the variable integration limits required on the simplex. Now my question is whether there is any general theory to evaluate rational functions on (subsets of) $S^{d-1}$ or in spherical coordinates in general. Rotating the system such that $y$ is axis aligned simplifies the numerator to a power of a cosine in one variable, which can be linearized and then the denominator is a linear combination of products of sines and cosines. But it definitely gets messy. Is there a simpler way?