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 $2.95$. 



evaluating an integral related to the volume of Hessenberg orthogonal matrices
Assuming that


$$I=\int\_0^{2\pi} \int\_0^{2\pi}\sqrt{9-\sin^2 \frac{\theta\_1 }{2} \sin^2 \frac{\theta\_2 }{2}}\mathrm{d}\theta\_1 \mathrm{d}\theta\_2$$


is correct,


$$I=3\int\_0^{2\pi} \int\_0^{2\pi}\sqrt{1-\frac19 \sin^2 \frac{\theta\_1 }{2} \sin^2 \frac{\theta\_2 }{2}}\mathrm{d}\theta\_1 \mathrm{d}\theta\_2$$


then,


$$I=12\int\_0^{2\pi} \int\_0^{\frac{\pi}{2}}\sqrt{1-\frac19 \sin^2 \theta\_1 \sin^2 \frac{\theta\_2 }{2}}\mathrm{d}\theta\_1 \mathrm{d}\theta\_2$$


$$I=12\int\_0^{2\pi}E\left(\frac19 \sin^2 \frac{\theta\_2 }{2}\right)\mathrm{d}\theta\_2$$


($E(m)$ is the complete elliptic integral of the second kind, with parameter $m$; for the Maple people, what you have is $E(k)$ where $k^2=m$)


$$I=48\int\_0^{\frac{\pi}{2}}E\left(\frac19 \sin^2 \theta\_2\right)\mathrm{d}\theta\_2$$


and letting $m=\sin^2 \theta\_2$,


$$I=24\int\_0^1 \frac1{\sqrt{m}\sqrt{1-m}} E\left(\frac{m}{9}\right)\mathrm{d}m$$


which *Mathematica* evaluates to


$$12\pi^2 {}\_3 F\_2\left(-\frac12,\frac12,\frac12 ; 1,1 ; \frac19\right)$$


where ${}\_3 F\_2$ is a hypergeometric function; further "simplification" can be done using the formula [here](http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric3F2/03/08/04/02/01/06/0001/).


