Understanding the intrinsic cosmic long gamma-ray burst (GRB) rate is essential in many aspects of astrophysics and cosmology, such as revealing the connection between GRBs, supernovae (SNe), and stellar evolution. Swift, a multi-wavelength space telescope, is quickly expanding the GRB category by observing hundreds of GRBs and their redshifts. However, it remains difficult to determine the intrinsic GRB rate due to the complex trigger algorithm adopted by Swift. Current studies of the GRB rate usually approximate the Swift trigger algorithm by a single detection threshold. Nevertheless, unlike the previously flown GRB instruments, Swift has over 500 trigger criteria based on count rates and additional thresholds for localization. To investigate possible systematic biases and further explore the intrinsic GRB rate as a function of redshift and the GRB luminosity function, we adopt a Monte Carlo approach by simulating all trigger criteria used by Swift. A precise estimation of the intrinsic GRB rate is important to reveal the GRB origins and their relation to the black-hole forming SNe. Additionally, the GRB rate at high redshifts provides a strong probe of the star formation history in the early universe, which is hard to measure directly through other methods.