For a typical hyperbolic tangential buffering packaging system, the approximate analytical solution of the system was discussed under the condition of drop impact. The hyperbolic tangent system was simplified to the cubic-quintic nonlinear system, and the dimensionless dynamic equation was obtained by introducing dimensionless parameters. The first-order, second-order and third-order approximate analytical solutions of the system response were obtained by the Newton-harmonic balancing method. Compared with the fourth-order Runge-Kutta numerical solution, the example analysis showed that the third-order approximate solution of the Newton-harmonic balancing method was the closest to the Runge-Kutta numerical solution, and the relative errors of the maximum displacement response, maximum acceleration response and dropping shock duration were controlled within 1%. A new approximate analysis method was provided for the drop impact response analysis of hyperbolic tangential nonlinear packaging system.