Consider the following integral, $$ \int_0^{2\pi} \int_0^{2\pi} (9- \sin^2 \frac{\theta_1 }{2} \sin^2 \frac{\theta_2 }{2})^{1/2} d\theta_1 d\theta_2.$$ 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.