Integrating by parts Maxwell Lagrangian

[Lapidus]

I attached a file that shows the free EM action integral and how it can be rewritten. I would like to know how to go from the first line to the second. I have to integrate by parts somehow, and I know surface terms get thrown out, but I do not know how the indices of the gauge fields should be handled.

thanks for any help

 

[Lapidus]

Has then perhaps anybody seen a text were it is derived?

I got -A∂∂A + A∂∂A, and both terms upper indices for A∂ and lower indices for ∂A. But what now? How to get to the last line?