  Simple inequalities are established for integrals of the type $\int_0^x
\mathrm{e}^{-\gamma t} t^{-\nu} \mathbf{L}_\nu(t)\,\mathrm{d}t$, where $x>0$,
$0\leq\gamma<1$, $\nu>-\frac{3}{2}$ and $\mathbf{L}_{\nu}(x)$ is the modified
Struve function of the first kind. In most cases, these inequalities are tight
in certain limits. As a consequence we deduce a tight double inequality,
involving the modified Struve function $\mathbf{L}_{\nu}(x)$, for a generalized
hypergeometric function.
