The computation of 3-D magnetic fields is a demanding task in the analysis of electrical machines and other electromagnetic devices. In this context, integral field calculation provides a smooth solution, high precision and resolution, "on-demand"-calculation, and an origin-based formulation of the magnetic field and the magnetic vector potential. However, conventional elliptic methods lead to huge parallelizable computing efforts and significant errors. In this article, a 3-D generic current-carrying arc segment with rectangular cross section is studied. A new analytic formulation is proposed to speed up the computation of magnetic fields and reduce the error by more than three orders of magnitude. In addition, the proposed magnetic vector potential expression has a similar accuracy as numerical integration. In fact, a significant reduction of the error level has been showcased clearly with respect to the existing approaches. This article is promising for improving the design methodology and optimization of large superconducting dipole magnets or arched end-winding geometries of large electrical machines.