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We study cubature formulas for d-dimensional integrals with an arbitrarysymmetric weight function of tensor product form. We present a constructionthat yields a high polynomial exactness: for fixed degree l=5 or l=7 and largedimension, the number of knots is only slightly larger than the lower bound ofM\"oller and much smaller compared to the known constructions.  We also show, for any odd degree l=2k+1, that the minimal number of points isalmost independent of the weight function. This is also true for theintegration over the (Euclidean) sphere.

Cubature formulas for symmetric measures in higher dimensions with few points