Fractional-order PID (FOPID) controllers have gained increasing interest in control theory in recent years, mainly to improve the performance and stability of complex systems. FOPID controllers are a generalization of classical PID controllers that have both integral and derivative orders of fractional order. This means that instead of having three tuning parameters as in classical PID, there are two additional degrees of freedom to achieve the control objectives. At the same time, the structure of the PID controller, so valued in industrial applications, is retained. A preferred method for designing FOPID controllers is the use of optimization algorithms. Different objective functions are used, such as the integral of the square error (ISE), the integral of the absolute error (IAE), the integral of the time-weighted absolute error (ITAE) or the integral of the time-weighted square error (ITSE), which must be minimized in order to adjust the five unknown parameters of the FOPID controller. There are several papers discussing different optimization methods and objective functions for different applications, but there are no general recommendations to help engineers choose the right method for their design. The present research compares three of the most well-known nature-inspired optimization techniques from the point of view of a FOPID controller design tool, using all the above enumerated objective functions. The analysis is realized both for processes with large time constants (such as thermal processes) and for dynamic systems (such as mechatronic processes), presenting the advantages and disadvantages of each method.