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We introduce a family of high-order time semi-discretizations for semilinearwave equations of Klein--Gordon type with arbitrary smooth nonlinerities thatare uniformly accurate in the non-relativistic limit where the speed of lightgoes to infinity. Our schemes do not require pre-computations that are specificto the nonlinearity and have no restrictions in step size. Instead, they relyupon a general oscillatory quadrature rule developed in a previous paper(Mohamad and Oliver, arXiv:1909.04616).

High-order uniformly accurate time integrators for semilinear wave equations of Klein-Gordon type in the non-relativistic limit