Find a limit of a specific n-dimensional integral as n approaches infinity

Question

I can show ( empirically, by Monte Carlo simulation ) that the $\displaystyle n$-dimensional integral of $\displaystyle\frac{1}{\pi^n\prod^{n}\left(1 + x_{i}^{2}\right)}$ over the region which meets $\displaystyle\frac{\sum^{n}x_{i}}{\,\sqrt{\,\sum^{n}x_{i}^{2}\,}\,} < X$ has a relatively simple limit ( a function of $\displaystyle X$ ), but I cannot figure out: what is the analytic form of this limit ? Note: $\displaystyle\pi$ is the usual $\displaystyle 3.1415\ldots$ constant, and the integral computes the CDF of a random variable when sampling from Cauchy distribution. Also note: the region of integration is an n-dimensional solid 'cone' with vertex at the origin and the (1,1,...1) axis. Finally, the integral over the whole space is clearly equal to 1, and the convergence to a limit appears to be very fast.