I've seen in couple sources that the gauge invariant Lagrangian for the Dirac field being written as follows:
$$\mathcal{L} = \frac{i}{2}[\bar{\psi}\gamma^{\mu}D_{\mu}\psi-(\bar{D}_{\mu}\bar{\psi})\gamma^{\mu}\psi]-m\bar{\psi}\psi.\tag{1}$$
I don't understand where the second term in the bracket, $(\bar{D}_{\mu}\bar{\psi})\gamma^{\mu}\psi$, comes from. One sees in text books that the interaction of the Dirac field with the gauge field is derived through the following Dirac Lagrangian (by changing $\partial_{\mu}$ to $D_{\mu}$):
$$\mathcal{L} = i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-m\bar{\psi}\psi.\tag{2}$$
So, I don't understand the source of the second term in the first Lagrangian.
