Let $\square_n=\{(x_1,\dots,x_n): 0\leq x_i\leq1,\, \forall i\}$ be an $n$-dimensional unit hypercube, and let $\Delta_n=\{(u_1,\dots,u_n):u_1+\cdots+u_n\leq\frac{\pi}2,\, u_i\geq0,\, \forall i\}$ be $n$-simplex.

Also, let $E_{2m}$ be *secant numbers* (even-index *Euler numbers*) given by
$$\sum_{m\geq0}E_{2m}\frac{y^{2m}}{(2m)!}.$$

The following has been experimentally tested.

Question.What is the transformation that makes this equality possible? $$\int_{\square_{2m+1}}\frac{d\mathbf{x}}{1+x_1^2\cdots x_{2m+1}^2}= \frac{(2m+1)E_{2m}}2\int_{\Delta_{2m+1}}d\mathbf{u}$$