The total amount of mental activity applied to working memory at a given point in time is called cognitive load, which is an important factor in various activities in daily life. We have proposed new feature quantities that reflect the time-series changes in the power of typical frequency bands in electroencephalogram (EEG) for use in examining the relationship between brain activity and behavior under cognitive load. We also measured heart rate variability (HRV) and spontaneous skin conductance responses (SCR) to examine functional associations among brain activity, autonomic activity, and behavior under cognitive load. Additionally, we applied our machine learning model previously developed using EEG to the estimation of arousal level to interpret the brain-autonomic-behavior functional association under cognitive load. Experimental data from 12 healthy undergraduate students showed that participants with higher levels of infra-slow fluctuations of alpha power have more cognitive resources and thus can process information under cognitive load more efficiently. In addition, HRV reflecting parasympathetic activity correlated with task accuracy. The arousal level estimated using our machine learning model showed its robust relationship with EEG. Despite the limitation of the sample size, the results of this pilot study suggest that the information processing efficiency of the brain under cognitive load is reflected by time-series fluctuations in EEG, which are associated with an individual's task performance. These findings can contribute to the evaluation of the internal state of humans associated with cognitive load and the prediction of human behaviors in various situations under cognitive load.