electromagnetism - Is $F_{munu}F^{munu}$ the only possible gauge-invariant Lagrangian for the electromagnetic field?

Maxwell's equations can be derived from a Lagrangian formulation using the Lagrangian term (modulo some constants)

$$\mathcal L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A_\mu J^\mu.$$ Focusing on the free term for the moment, I've seen mentioned (though I can't find a source right now) that the $F^2$ term is the only possible gauge-invariant Lagrangian for electromagnetism.

Is this the case? How can we prove it?