Recently, a mixed-radix decimation in frequency (DIF) fast MDCT algorithm only for the mixed-radix decompositions or composite lengths N = 3^m.2, m>0, has been proposed in ShuBao. An improved mixed-radix DIF fast MDCT algorithm both in terms of the regularity and computational complexity is described. Based on observed simple algebraic identities in the original proposed algorithm ShuBao, new formulas are derived resulting in a very regular computational structure. Consequently, the number of arithmetic operations is reduced significantly. Moreover, the improved algorithm is extended to all composite lengths N = 3^m.2^p, m, p>0. The improved algorithm defines new sparse matrix factorizations of the MDCT matrix for the composite lengths N = 3^m.2^p, m, p>0, and finally it provides new implementations of the forward/backward MDCT in MPEG-1/2 layer III (MP3) audio coding standard.