<p>We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based formulas are based on the inverse Laplace transform or the Fourier transform. We show these expressions are essentially equivalent in terms of complex integrals and choose the former as a starting point. By choosing a simple integral path, we derive an integral expression mentioned above. Then, we can easily apply the double-exponential formula and the Gauss–Legendre formula, which have explicit error bounds. As numerical experiments show, the proposed formula can be comparable with the existing formulas.</p>