  Simple upper and lower bounds are established for the integral
$\int_0^x\mathrm{e}^{-\beta t}t^\nu \mathbf{L}_\nu(t)\,\mathrm{d}t$, where
$x>0$, $\nu>-1$, $0<\beta<1$ and $\mathbf{L}_\nu(x)$ is the modified Struve
function of the first kind. These bounds complement and improve on existing
results, through either sharper bounds or increased ranges of validity. In
deriving our bounds, we obtain some monotonicity results and inequalities for
products of the modified Struve function of the first kind and the modified
Bessel function of the second kind $K_{\nu}(x)$, as well as a new bound for the
ratio $\mathbf{L}_{\nu}(x)/\mathbf{L}_{\nu-1}(x)$.
