  Three-centre nuclear attraction integrals, which arise in density functional
and \textit{ab initio} calculations, are one of the most time-consuming
computations involved in molecular electronic structure calculations. Even for
relatively small systems, millions of these laborious calculations need to be
executed. Highly efficient and accurate methods for evaluating molecular
integrals are therefore all the more vital in order to perform the calculations
necessary for large systems. When using a basis set of $B$ functions, an
analytical expression for the three-centre nuclear attraction integrals can be
derived via the Fourier transform method. However, due to the presence of the
highly oscillatory semi-infinite spherical Bessel integral, the analytical
expression still remains problematic. By applying the $S$ transformation, the
spherical Bessel integral can be converted into a much more favorable sine
integral. In the present work, we then apply two types of double exponential
transformations to the resulting sine integral, which leads to a highly
efficient and accurate quadrature formulae. This method facilitates the
approximation of the molecular integrals to a high predetermined accuracy,
while still keeping the calculation times low. The fast convergence properties
analyzed in the numerical section illustrate the advantages of the method.
