The 2nd time... there's a video...
There is NO video of the second swearing in - it was a small gathering of White House aides, along with a handful of reporters (four, I believe) and no audio or video was taken. A single still photo of the event was released.
I am not claiming a grand secret conspiracy - I think they just fumbled their words the first time and decided to have a “re-do” to be certain....but, as with so many things related to Obama, it was an unusual, and secretive, event.