  We study the work statistics of a periodically-driven integrable closed
quantum system, addressing in particular the role played by the presence of a
quantum critical point. Taking the example of a one-dimensional transverse
Ising model in the presence of a spatially homogeneous but periodically
time-varying transverse field of frequency $\omega_0$, we arrive at the
characteristic cumulant generating function $G(u)$, which is then used to
calculate the work distribution function $P(W)$. By applying the Floquet theory
we show that, in the infinite time limit, $P(W)$ converges, starting from the
initial ground state, towards an asymptotic steady state value whose small-$W$
behaviour depends only on the properties of the small-wave-vector modes and on
a few important ingredients: the time-averaged value of the transverse field,
$h_0$, the initial transverse field, $h_{\rm i}$, and the equilibrium quantum
critical point $h_c$, which we find to generate a sequence of non-equilibrium
critical points $h_{*l}=h_c+l\omega_0/2$, with $l$ integer. When $h_{\rm i}\neq
h_c$, we find a "universal" edge singularity in $P(W)$ at a threshold value of
$W_{\rm th}=2|h_{\rm i}-h_c|$ which is entirely determined by $h_{\rm i}$. The
form of that singularity --- Dirac delta derivative or square root --- depends
on $h_0$ being or not at a non-equilibrium critical point $h_{*l}$. On the
contrary, when $h_{\rm i}=h_c$, $G(u)$ decays as a power-law for large $u$,
leading to different types of edge singularity at $W_{\rm th}=0$. Generalizing
our calculations to the case in which we initialize the system in a finite
temperature density matrix, the irreversible entropy generated by the periodic
driving is also shown to reach a steady state value in the infinite time limit.
