Abstract: Accurate and efficient integration of the equations of motion is indispensable for molecular dynamics (MD) simulations. Despite the massive use of the conventional leapfrog (LF) integrator in modern computational tools within the framework of MD propagation, further development for better performance is still possible. The alternative version of LF in the middle thermostat scheme (LF-middle) achieves a higher order of accuracy and efficiency and maintains stable dynamics even with the integration time stepsize extended by several folds. In this work, we perform a benchmark test of the two integrators (LF and LF-middle) in extensive conventional and enhanced sampling simulations, aiming at quantifying the time-stepsize-induced variations of global properties (e.g., detailed potential energy terms) as well as of local observables (e.g., free energy changes or bond-lengths) in practical simulations of complex systems. The test set is composed of six chemically and biologically relevant systems, including the conformational change of dihedral flipping in the N-methylacetamide and an AT (Adenine-Thymine) tract, the intra-molecular proton transfer inside malonaldehyde, the binding free energy calculations of benzene and phenol targeting T4 lysozyme L99A, the hydroxyl bond variations in ethaline deep eutectic solvent, and the potential energy of the blue-light using flavin photoreceptor. It is observed that the time-step-induced error is smaller for the LF-middle scheme. The outperformance of LF-middle over the conventional LF integrator is much more significant for global properties than local observables. Overall, the current work demonstrates that the LF-middle scheme should be preferably applied to obtain accurate thermodynamics in the simulation of practical chemical and biological systems.