Application of the acoustic Fourier series boundary element method (FBEM) for axisymmetric structures with non-axisymmetric boundary conditions is very attractive, because of the significantly reduced computational cost compared to application of the normal 3-dimensional BEM. An efficient implementation of the acoustic FBEM however, is far from straightforward. In the FBEM formulation some integrals over the angle of revolution arise, which need to be calculated for every Fourier term. The evaluation time for these integrals determines the total computational cost and therefore the efficiency of the FBEM. The integrals were formerly treated for each Fourier term separately with trapezoidal or Gaussian quadrature. A new method was developed to calculate these integrals using Fast Fourier Transform techniques. The advantage of this new method is that the integrals are computed simultaneously for all Fourier terms in the boundary element formulation. In this paper, the improved efficiency of the method compared to a Gaussian quadrature based integration algorithm is illustrated by some simplified example calculations. In real-life applications with generally complex geometry and boundary conditions, the advantage of the new method is exploited to its full extend.