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This article considers the computation of the matrix exponential eA{{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore, it is necessary to truncate the infinite interval. In this article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates eA{{\rm{e}}}^{A} with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes eA{{\rm{e}}}^{A} based on the first one with automatic selection of the mesh size depending on the given error tolerance.