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Consider the regions $r1 and $r2 defined as follows,

$r1 =HalfSpace[{Sqrt[3], -1, -((3 \[Epsilon] )/Sqrt[12 + \[Epsilon]^2]), (Sqrt[3] \[Epsilon] )/Sqrt[12 + \[Epsilon]^2]}, 0];

$r2 = HalfSpace[{-Sqrt[3] Sqrt[12 + \[Epsilon]^2], -Sqrt[
    12 + \[Epsilon]^2], 3 \[Epsilon], Sqrt[3] \[Epsilon]}, 0];

where \[Epsilon] is a parameter defined in the interval $[0,\pi/2)$.

How can I compute the following integral?

Integrate[Exp[-x^2 - y^2 - u^2 - v^2], {x, y, u, v} \[Element] RegionIntersection[$r1, $r2], Assumptions -> {\[Pi]/2 > \[Epsilon] >= 0}]

I tried using Reduce only to get a extremely complicated expression containing Root that MMA is also incapable to integrate. Using /._Equal -> False did not help.

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    $\begingroup$ First, your function is symmetric in all variables. Second, the origin is on the boundary of the Half Space. Therefore, the integral is independent of the direction of the normal. You can get the value by e.g. setting epsilon=0: Pi^2/2 $\endgroup$ – Daniel Huber Mar 7 at 15:53
  • $\begingroup$ @DanielHuber, yes, you're right. I'll need to edit my question due to the fact that when coming up with a MWE I apparently oversimplified the original problem. Sorry about that. $\endgroup$ – David Mar 7 at 16:10

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