In maritime engineering, ensuring vessel stability remains a paramount concern. This study investigates the hydrodynamic response of Magnus anti-rolling devices, modeled as swinging or slewing rotating cylinders, under a ship's rolling motion. Through numerical simulations using the overset mesh technique and large eddy simulation, we analyze various parameters, including rolling angles, rotating speeds, and swinging amplitudes. Our findings highlight the importance of considering the ship's degree of freedom as substantial ship rolling significantly affects hydrodynamic coefficients on the rotating cylinder. We observe interesting dynamics during slewing motion, with the cylinder forming a spiral tip vortex. Optimizing the cylinder's rotating speed enhances the lift-to-drag ratio, particularly for small rolling angles. Furthermore, the effective lift generated during swinging motion is lower than during slewing motion, emphasizing the need to optimize the swinging amplitude, which is recommended to be no less than 170 degrees. These insights advance our understanding of Magnus anti-rolling devices and offer practical guidance for improving vessel stability in complex maritime environments.