In this paper, we develop low complexity and stable bootstrap procedures for FastICA estimators. Our bootstrapping techniques allow for performing cost efficient and reliable bootstrap-based statistical inference in the ICA model. Performing statistical inference is needed to quantitatively assess the quality of the estimators and testing hypotheses on mixing coefficients in the ICA model. The developed bootstrap procedures stem from the fast and robust bootstrap (FRB) method , which is applicable for estimators that may be found as solutions to fixed-point (FP) equations. We first establish analytical results on the structure of the weighted covariance matrix involved in the FRB formulation. Then, we exploit our analytical results to compute the FRB replicas at drastically reduced cost. The developed enhanced FRB method (EFRB) for FastICA permits using bootstrap-based statistical inference in a variety of applications (e.g., EEG, fMRI) in which ICA is commonly applied. Such an approach has not been possible earlier due to incurred substantial computational efforts of the conventional bootstrap. Our simulation studies compare the complexity and numerical stability of the proposed methods with the conventional bootstrap method. We also provide an example of utilizing the developed bootstrapping techniques in identifying equipotential lines of the brain dipoles from electroencephalogram (EEG) recordings.