Starting from the basic equations describing the evolution of the carriers and photons inside a semiconductor optical amplifier (SOA), the equation governing pulse propagation in the SOA is derived. By employing homotopy analysis method (HAM), a series solution for the output pulse by the SOA is obtained, which can effectively characterize the temporal features of the nonlinear process during the pulse propagation inside the SOA. Moreover, the analytical solution is compared with numerical simulations with a good agreement. The theoretical results will benefit the future analysis of other problems related to the pulse propagation in the SOA.