Discretizing wave equations in space typically results in large systems of second-order differential equations. For the time integration of such systems, classical Runge-Kutta-Nyström integrators and their extended variants have been widely used. While effective for small or non-stiff problems, these methods often suffer from stability issues and inefficiency when applied to large, stiff or highly oscillatory systems. To overcome these limitations, we develop and analyze a new class of time integration methods, called exponential Nyström (expN) methods. These methods allow significantly larger time steps without sacrificing accuracy. We establish convergence results up to fifth-order accuracy within the framework of strongly continuous semigroups, with error bounds independent of the stiffness or high frequencies of the system. Our numerical experiments demonstrate that the proposed expN methods outperform existing Nyström-type integrators in both efficiency and accuracy.