Consider the following integral,
$$
{1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right)
            \sin^{2}\left(\theta_{2} \over 2\right)\,}
\,{\rm d}\theta_{1}\,d\theta_{2}
$$
This integral comes up in computing the volume of $3$-dimensional special orthogonal matrices of Hessenberg form, i.e., the bottom left entry is $0$. Mathematica isn't able to produce close form solution. Numerically it's about $2.95$.