I can’t reproduce your problem (Julia 0.5 on MacOS).
julia> hcubature(x -> x[1] * (0.3 - 5*x[1]*sin(x[2])).^2, [0,0], [0.2,2π], reltol=1e-6, abstol=1e-4)
(0.042725660088821185,1.348693715190561e-5)
julia> hcubature(x -> x[1] * (0.3 - 5*x[1]*cos(x[2])).^2, [0,0], [0.2,2π], reltol=1e-6, abstol=1e-4)
(0.042725660088821185,1.3486937151908646e-5)
julia> freq_range = linspace(-0.5, +0.5, 4);
julia> field_sin = p -> 5 * p[1] * sin(p[2]);
julia> field_cos = p -> 5 * p[1] * cos(p[2]);
julia> test_cuba_method(field_cos, 1e-6, 1e-4, freq_range, :hcuba)
([0.5,0.277778,0.277778,0.5],[8.94878e-5,8.83078e-5,8.71277e-5,8.59476e-5])
julia> test_cuba_method(field_sin, 1e-6, 1e-4, freq_range, :hcuba)
([0.5,0.277778,0.277778,0.5],[8.94878e-5,8.83078e-5,8.71277e-5,8.59476e-5])
i.e. I appear to be getting the same results for cos and sin, both with your posted test_cuba_method code and with a simplified 1-output case.