To answer your question, Hussein's long form birth certificate was made available to the public in a downloadable pdf format on the White House gov website April 27, 2011 by his former assistant and senior advisor in strategy and communications, Dan Pfeiffer.
I don't want this to be a slide, but was curious too. I doubt if they had access to an original.
https://obamawhitehouse.archives.gov/blog/2011/04/27/president-obamas-long-form-birth-certificate