I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int_D f(x,y)dxdy$ numerically. After much searching, the best methods I've been able to find have been the simple quadrature rules in Abramowitz and Stegun that sample $f$ at up to 21 points. What work has been done since? In particular, I'm interested in rules that allow sampling at more than 21 points. One reference indicates that finding optimal quadrature rules is a hard problem, but it seems to me that something better must have been published in the last 50 years.

A couple of references suggest integrating over various domains by triangulating them and using numerical integrals over the triangles. Is this the preferred method for a disk?

(I'm trying to improve code that has been implemented using quasi-monte carlo. It seems to me that we could do much better using the knowledge that $f$ is real-analytic and probably well approximated by polynomials.)

Update: I can't easily say exactly what the functions $f$ are as they're the messy result of a chain of computations. I can say that qualitatively it's like a gaussian with a central hump, fast decay, though not exact rotational symmetry. I do have a pretty good handle on how big the hump is and where it's centred. All variations on this might happen and some are easy to dispense with: eg. the hump might be situated well outside the disk so I know the integral is nearly zero. Or the hump may be very wide in which case the integrand is almost constant. Sometimes the hump is contained well within the disk in which case I can switch to more efficient quadrature over the (approximate) support of $f$ rather than the disk. But having said all that, I'd still like to see some general gaussian quadratures rules for the disk that would apply to integrating any function over the disk that is well approximated by a polynomial.

Update2: After much web searching I found some Fortran code to do what I want (and more) and a reference to a book by Arthur Stroud, *Approximate Calculation of Multiple Integrals*. It seems as that this work from 1971 is the state of the art.