  We consider the asymptotic expansion of the generalised exponential integral
involving the Mittag-Leffler function introduced recently by Mainardi and
Masina [{\it Fract. Calc. Appl. Anal.} {\bf 21} (2018) 1156--1169]. We extend
the definition of this function using the two-parameter Mittag-Leffler
function. The expansions of the similarly extended sine and cosine integrals
are also discussed. Numerical examples are presented to illustrate the accuracy
of each type of expansion obtained.
