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Pulsar Timing Array (PTA) experiments are expected to be sensitive to gravitational waves (GWs) emitted by individual supermassive black hole binaries (SMBHBs) inspiralling along eccentric orbits. We compare the computational cost of different methods of computing the PTA signals induced by relativistic eccentric SMBHBs, namely approximate analytic expressions, Fourier series expansion, post-circular expansion, and numerical integration. We show that the fastest method for evaluating PTA signals is by using the approximate analytic expressions, which provides up to ~50 times improvement in computational speed over the alternative methods. We investigate the accuracy of the approximate analytic expressions by employing a mismatch metric valid for PTA signals. We show that this method is accurate within the region of the binary parameter space that is of interest to PTA experiments. We introduce a spline-based method to further accelerate the PTA signal evaluations for narrowband PTA datasets. The efficient methods for computing the eccentric SMBHB-induced PTA signals were implemented in the GWecc.jl package and can be readily accessed from the popular ENTERPRISE package to search for such signals in PTA datasets. Further, we simplify the eccentric SMBHB PTA signal expression for the case of a single-pulsar search and demonstrate our computationally efficient methods by performing a single-pulsar search in the 12.5-year NANOGrav narrowband dataset of PSR J1909-3744 using the simplified expression. These results will be crucial for searching for eccentric SMBHBs in large PTA datasets.