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We present a novel direct integral pseudospectral (PS) method (a direct IPS method) for solving a class of continuous-time infinite-horizon optimal control problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal control problems (FHOCs) in their integral forms by means of certain parametric mappings, which are then approximated by finite-dimensional nonlinear programming problems (NLPs) through rational collocations based on Gegenbauer polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes the interplay between the parametric maps, barycentric rational collocations based on Gegenbauer polynomials and GGR points, and the convergence properties of the collocated solutions for IHOCs. Some novel formulas for the construction of the rational interpolation weights and the GGR-based integration and differentiation matrices in barycentric-trigonometric forms are derived. A rigorous study on the error and convergence of the proposed method is presented. A stability analysis based on the Lebesgue constant for GGR-based rational interpolation is investigated. Two easy-to-implement pseudocodes of computational algorithms for computing the barycentric-trigonometric rational weights are described. Two illustrative test examples are presented to support the theoretical results. We show that the proposed collocation method leveraged with a fast and accurate NLP solver converges exponentially to near-optimal approximations for a coarse collocation mesh grid size. The paper also shows that typical direct spectral/PS- and IPS-methods based on classical Jacobi polynomials and certain parametric maps usually diverge as the number of collocation points grow large, if the computations are carried out using floating-point arithmetic and the discretizations use a single mesh grid whether they are of Gauss/Gauss-Radau (GR) type or equally-spaced.