Five reliable integration architectures are adopted in this paper, namely the unified Riccati equation expansion, the modified simple equation, the extended simplest equation, the generalized sub-ODE, and the new extended auxiliary equation approaches for the extraction of optical soliton solutions and other solutions in birefringent fibers without four-wave mixing (FWM) terms for the Biswas-Arshed model. Bright soliton, kink and dark soliton, singular soliton, hyperbolic wave, a periodic wave, elliptic function solutions of Wierestrass and Jacobian, and other solutions have been derived as a result. The conditions of their existence are also revealed. Also, the results obtained are compared with the well-known ones listed for the Biswas-Arshed model in birefringent fibers without FWM. The amplitude and intensity diagrams of some of the attained solutions in this study are portrayed by providing the appropriate values of the parameters included in the solutions for physical description.