This is only a partial answer. The Beukers-Kolk-Calabi change of variables
$$x_1=\frac{\sin{u_1}}{\cos{u_2}},\;\;x_2=\frac{\sin{u_2}}{\cos{u_3}},\ldots,
\;x_{n-1}=\frac{\sin{u_{n-1}}}{\cos{u_n}},\;\;x_n=\frac{\sin{u_n}}{\cos{u_1}}$$
has the Jacobian
$$\frac{\partial(x_1,\ldots,x_n)}{\partial(u_1,\ldots,u_n)}=
1-(-1)^n\,x^2_1x^2_2\cdots x^2_n.$$ 
Therefore you integral is a volume of the polytope $\delta_{2m+1}=\left \{(u_1,\ldots,u_{2m+1}): u_i\ge 0,\; u_i+u_{i+1}\le \pi/2
\right \}$. It remains to relate the volumes of the polytope $\delta_{2m+1}$ and $(2m+1)$-simplex $\Delta_{2m+1}$. See https://arxiv.org/abs/math/0101168