I have a set of parameters $x_1, \dots x_4 \in (0,1)$. I want to find the volume of the subregion of this 4d cube satisfying $$\sin^{-1}(x_1)+ \dots +\sin^{-1}(x_4) > \frac{3\pi}{2}$$
I've tried using a simple
NIntegrate[ If[ArcSin[x1] + ArcSin[x2] + ArcSin[x3] + ArcSin[x4] > (3 Pi)/2, 1,
0], {x4, 0, 1}, {x3, 0, 1}, {x2, 0, 1}, {x1, 0, 1}]
However I get numerical lamentations. If I solve for one of the $x_i$ and put $x_i = \sin(\frac{3\pi}{2}-\sum_{j\neq i} \sin^{-1}{x_j})$ as a lower limit in its integral, I lose information by taking the sine ($\sin(x)$ is two-to-one for $x<2\pi$) and the $x_i$ integral can move beyond its range of validity.
Is there an elegant way to this without many If/else statements?