  We present a comprehensive first principles electronic structure study of the
magnetoelastic and magnetostrictive properties in the Co-based Co$_2$XAl (X =
V, Ti, Cr, Mn, Fe) full Heusler compounds. In addition to the commonly used
total energy approach, we employ torque method to calculate the magnetoelastic
tensor elements. We show that the torque based methods are in general
computationally more efficient, and allow to unveil the atomic- and
orbital-contributions to the magnetoelastic constants in an exact manner, as
opposed to the conventional approaches based on second order perturbation with
respect to the spin-orbit coupling. The magnetostriction constants are in good
agreement with available experimental data. The results reveal that the main
contribution to the magnetostriction constants, $\lambda_{100}$ and
$\lambda_{111}$, arises primarily from the strained-induced modulation of the
$\langle d_{x^2-y^2}|\hat{L}_z|d_{xy}\rangle$ and $\langle
d_{z^2}|\hat{L}_x|d_{yz}\rangle$ spin orbit coupling matrix elements,
respectively, of the Co atoms.
