There is no need to investigate.
We already know the answer(s).
1) he was a dual national, by his own admission = not a Natural Born Citizen
2) he was, in fact, ineligible for the office of President by his own admission.
Simultaneously, I grant that "even though what he has claimed" clearly suggests (in my best understanding) that he is not a Natural Born Citizen. That nonetheless would need to be adjudicated through the legal system. Perhaps a tribunal can get to the right answer expeditiously.
The many legal cases that might have tried to see that question adjudicated on the merits were dismissed, usually on the basis of a lack of standing.
Dick "DS" Cheney could have had the question adjudicated on the basis of his presiding over the 12/15/2008 convocation of the presidential electors, but as that could easily have gone in a way averse to his personal interests, he didn't "bother."
Orly Taitz had a venue with the Mississippi Democrats for which the Hawai'i Department of Health had prepared an invalid Verification, which could well have led to the case being decided on the merits, but Taitz was either too incompetent as a lawyer, or she never intended to see the question settled adverse to Barry's interests.