Attitude and Heading Reference Systems (AHRSs) are broadly applied wherever reliable orientation and motion sensing is required. In this paper, we present an improved Cubature Kalman Filter (CKF) with lower computational cost while maintaining estimation accuracy, which is named "Kaisoku Cubature Kalman Filter (KCKF)". The computationally efficient equations of the KCKF are derived by simplifying those of the CKF, while preserving equivalent mathematical relations. The lightweight prediction equations in the KCKF are derived by expanding the summation terms in the CKF and simplifying the result. This paper shows that the KCKF requires fewer floating-point operations (FLOPs) than the CKF. The controlled experimental results show that the KCKF reduces the computation time by approximately 19% compared to the CKF on a high-performance computer, whereas the KCKF reduces the computation time by approximately 15% compared to the CKF on a low-cost single-board computer. In addition, the KCKF maintains the attitude estimation accuracy of the CKF.