EFFICIENT CUBATURE RULES

67 new cubature rules are found for three standard multi-dimensional integrals with spherically symmetric regions and weight functions using direct search with a numerical zero-finder. 63 of the new rules have fewer integration points than known rules of the same degree, and 20 are within three points of Moller's lower bound. Most have all positive coefficients, and most have some symmetry, including some supported by one or two concentric spheres. They include degree-7 formulas for the integration over the sphere and Gaussian-weighted integrals over the entire space, each in 6 and 7 dimensions, with 127 and 183 points, respectively.