Focusing on the problem that the filtering precision of the traditional cubature Kalman filter(CKF) decreases under the condition that a system model mismatches and sudden changes of state happen, an algorithm called reduced high-degree strong tracking cubature Kalman filter(RHSTCKF) is proposed, which combines the strong tracking filter with high-degree cubature Kalman filter. The algorithm has higer filtering precision than the traditional CKF, simplifying caculation steps by using the characteristics of the filtering model, and strengthening adaptivity and the ability of sudden changes by introducing the suboptimal multiple fading factor. The proposed algorithm is applied to the SINS/GPS integrated navigation system, and the simulation results show that the RHSTCKF can accurately estimate true value although the state has sudden changes, restrain the interference when filters have abnormal states, obviously improve the filter performance, and enhance the adaptivity and the positional accuracy of the integrated navigation system.