Summary: This paper presents a new algorithm for the rapid and accurate calculation of the complex exponential integral associated with the mutual impedance of sinusoidal basis and testing functions in the method of moments. The new algorithm uses Leibniz's theorem to calculate Taylor series expansions of the integral instead of integrating expansions of the integrand as is often done. This results in an algorithm which is twice as fast as and is valid over a wider range than previous algorithms. This technique can be applied to many other integrals encountered in computational electromagnetics as well.