Are your integrals from -\infty to +\infty? You could try using a specialized quadrature rule, e.g. Gauss–Hermite quadrature would be reasonable here given the \exp(-h^2/2) weight factor. The GitHub - JuliaApproximation/FastGaussQuadrature.jl: Julia package for Gaussian quadrature package can generate such rules for you. (This is for a fixed quadrature order, so you’d have to do some experimentation to find a good quadrature order for your problem.)
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