Index notation with four-gradient

Question

Reading Schwarz's textbook on quantum field theory, early on he gives the Lagrangian $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_{\mu}J_{\mu}.$$ With $F^{\mu\nu}=(\partial_{\mu}A^{\nu}-\partial_{\nu}A^{\mu})$, he simplies the Lagrangian to $$\mathcal{L}=-\frac{1}{2}(\partial_{\mu}A_{\nu})^2+\frac{1}{2}(\partial_{\mu}A_{\mu})^2-A_{\mu}J_{\mu}.$$ I tried to do the working to get to this point, \begin{align*} \mathcal{L}&=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_{\mu}J_{\mu}\\ &=-\frac{1}{4}\left[\Big(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\Big)\Big(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\Big)\right]-A_{\mu}J_{\mu}\\ &=-\frac{1}{4}\left[\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu}-\partial_{\mu}A_{\nu}\partial_{\nu}A_{\mu}-\partial_{\nu}A_{\mu}\partial_{\mu}A_{\nu}+\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}\right]\\ &=-\frac{1}{2}\Big(\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu}\Big)+\frac{1}{4}\Big(\partial_{\mu}A_{\nu}\partial_{\nu}A_{\mu}+\partial_{\nu}A_{\mu}\partial_{\mu}A_{\nu}\Big)-A_{\mu}J_{\mu}. \end{align*} I can see how the first term is obviously $-1/2(\partial_{\mu}A_{\nu})^2$, but I can't see how the second term simplifies to $1/2(\partial_{\mu}A_{\mu})^2$. Do we just let $\mu \leftrightarrow \nu$, or something else?