Quasi-linear diffusion by cyclotron-resonant plasma waves is likely a key ingredient of the behavior of electrons in the Earth's radiation belts. Multidimensional dynamical simulations are under development, which require the diffusion coefficients to be evaluated quickly as well as accurately. The recently developed parallel propagation approximation replaces the integration over wavenormal distribution to a closed form expression, and can be quite accurate. However, it can also perform badly, especially for electron energy > or = 1 MeV. Here, the accuracy, justification, and limits of the approximation are explored, and an improved version is presented. It is based on a previously developed procedure for identifying wavenormal angles compatible with imposed cutoffs on the wave frequency distribution. Because it also requires evaluation at only a small number of points, it features computational efficiency comparable to the parallel propagation version, while preserving contributions from oblique waves and all relevant harmonic numbers. Detailed comparisons are presented using an established model of nightside chorus.