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A direct integration algorithm, based on double exponential-type quadrature rules, is presented for the efficient computation of the Sommerfeld integral tails, arising in the evaluation of multilayered Green's functions. The proposed scheme maintains the error controllable nature of the so-called partition-extrapolation methods, often used to tackle this problem, whereas it requires substantially reduced computational time. Moreover, the proposed method is very easy to implement, since the associated weights and abscissas can be precomputed. The overall behavior of the proposed method both in terms of accuracy and efficiency is demonstrated through a series of representative numerical experiments, where compared with one of the most proven methods available in the literature.