**7. Conclusions**

We reviewed various trigonometrically-fitted methods and implemented representatives on several examples of second order initial value problems. In most examples our method from [51] was superior to others except for the first example for which Gautschi's method performed better. The methods (32) and (66) due to Neta and Ford failed to converge for the last two examples and in the long-term integration of example 1. The method based on Runge-Kutta-Nyström due to Demba et al. [49] could not compete with our method based on Obrechkoff.

**Author Contributions:** Conceptualization, B.N.; Investigation, B.N.; Software, C.C.; Writing—original draft, B.N.; Writing—review & editing, C.C.

**Funding:** This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1A09917373).

**Acknowledgments:** The first author thanks the Applied Mathematics Department at the Naval Postgraduate School for hosting him during the years. The authors thank the referees for their valuable comments.

**Conflicts of Interest:** The authors declare no conflict of interest.
