We analytically study the exact solitary wave solutions of the perturbed nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index, which describes the propagation of pulses of various types in optical fiber. We apply three efficient and reliable schemes, specifically, the simple equation method, the (G'/G)-expansion method, and the new Kudryashov method. These approaches lead to a range of solitons and other solutions comprising of the bright solitons, dark solitons, singular solitons, periodic, rational, and exponential solutions. These solutions are also presented graphically. Furthermore, all obtained solutions are verified by symbolic computations.