The Gautschi-type exponential integrators, which are discussed in the talk, are proposed in [1] and [2] for oscillatory differential equations. These integrators are studied for second-order differential equations in which high-frequency oscillations are generated by a linear time- or solution-dependent part. The presented results show that the methods admit second-order error bounds which are independent of the product of the step-size with the frequencies. Methods with this property are called long-time-step methods in [3]. Applications for which long-time-step methods are intended include molecular dynamics. The appearing systems of ordinary differential equations often allow a splitting of the right-hand-side in fast forces, which generate the high frequencies, and slow forces. If the fast forces can be evaluated more cheaply than the full right-hand-side, long-time-step methods are superior to other integration schemes. Some results of numerical experiments are included.