Dear Habibur, it is difficult to code ODE integrators more efficiently than Matlab. Moreover, as it always happens, there is no best solution to this problem: the choice depends on your specific situation. Each Matlab basic function ode* is suited for a specific case and a specific level of accuracy, see

https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html

As a first point for efficiency, you should choose the function closer to your needs (in particular, *s functions for stiff problems). As a second, you should set the accuracy for adaptive solvers to your minimum required level, so as to avoid overly small integration steps, which slow down the computation.

Thank you for your answer. Roberto Ferretti