# computing fairly complicated integrals on the sphere

Assume $D$ is a diagonal matrix $2n\times 2n$. Let $J$ be the standard symplectic matrix, i.e.: $$J:=\begin{pmatrix} 0 &-\text{id}\\ \text{id} &0\end{pmatrix}.$$ I would like to compute the following integral: $$I(D):=\int_{\mathbb{S}^{2n-1}} \frac{\lvert -Dw+Jw\rvert}{(1+\langle w,Dw\rangle^2)^{\frac{2n+3}{4}}} d\sigma(w),$$ where as usual $\mathbb{S}^{2n-1}$ denotes the unit sphere in $\mathbb{R}^{2n}$ and $\sigma$ is its surface measure.

I am intrested in this integral as I would like to solve an equation of the kind: $$\text{Tr}(D)=I(D).$$