hello everyone,

I'm trying to compute a triple integral in mathematica (with NIntegrate), but it's not giving me a good answer. In fact it sometimes gives me a number close to 22, sometimes (if I increase working presicion) close to 90, etc (I'm actually expecting this integral to be less than 1). And everytime it also says that integral converges too slowly. However the function is positive, so I don't understand why is it so hard to compute. Are there tricks I could use to evaluate the integral precisely, in these cases?

I tried Monte Carlo method, and it gives me even more random answers...

Btw, the function does have blow-ups on the edges, but they are of the form , which is integrable.

Just in case someone interested, here's my function:

NIntegrate[ 2*1/(3!*\[Pi]^2) * r*(2 r*Sin[t]*(r^2 + z1^2 - 2*r*z1*Cos[t]))/((1 + r^2 z1^2 - 2 r z1 Cos[t])*Sqrt[((1 + r^2*Sin[t]^2 - r^2*Cos[t]^2)^2 + 4*r^4*Sin[t]^2*Cos[t]^2)]*Sqrt[1 - z1^2]*(1 - r^2)), {z1, -1, 1}, {r, 0, 1}, {t, 0, \[Pi]}, MaxRecursion -> 50, WorkingPrecision -> 200, PrecisionGoal -> 8]