The case includes the fact that he has admitted that he was a British Subject at birth. He cannot be a natural born citizen.
I am not a lawyer, but that appears to be a matter of some dispute. That statement appears to depend on whether the Founding Fathers depended on Emerich de Vattel's 1758 treatise, The Law of Nations or the Principles of Natural Law Applied to the Conduct and to the Affairs of Nations and of Sovereigns, as the source of their thinking. That document was written in French by a Swiss philosopher, and was not part of English common law. It has been cited frequently as part of the "just war" doctrine, but, AFAIK, has never been used as a precedent for a US Supreme Court decision. It was cited in the dissent in the case of The United States v. Wong Kim Ark, in 1898. The majority decision cited English common law as stating that, "every child born in England of alien parents was a natural-born subject," and held that to be the controlling precedent for US common law.
There is also the obvious precedent of Chester A. Arthur, the 21st President. There is some similarity to Obama's situation, since there was a dispute at the time of his election to the Vice-Presidency regarding his place of birth. There were rumors that he had actually been born in Ireland or Canada rather than in Fairfield, Vermont. The fact that his father was a British subject from Northern Ireland, and not naturalized as a US citizen until 1843, was undisputed and did not appear to be an issue to the electorate of the time.
“Rep. John A. Bingham commenting on Section 1992 said it means every human being born within the jurisdiction of the United States of parents not owing allegiance to any foreign sovereignty is, in the language of your Constitution itself, a natural born citizen. (Cong. Globe, 39th, 1st Sess., 1291 (1866))
Bingham subscribed to the same view as most everyone in Congress at the time that in order to be born a citizen of the United States one must be born within the allegiance of the Nation. Bingham had explained years earlier that to be born within the allegiance of the United States the parents, or more precisely, the father, must not owe allegiance to some other foreign sovereignty (remember the U.S. abandoned Englands natural allegiance doctrine). This of course, explains why emphasis of not owing allegiance to anyone else was the affect of being subject to the jurisdiction of the United States.”
Chester Artur hid his birthplace and parents citizenship from the electorate. He was a fraud like Obama.