As to the use of “so help me God” in the 2009 oath, and the clergy prayers in 2009, the panel (Judge Janice Rogers Brown, joined by Judge Douglas Ginsburg) held the challenge was moot; as to the challenge to similar elements in the 2013 and 2017 inaugurations, the Court found lack of standing, on non-redressability grounds. Judge Brett Kavanaugh concurs in the judgment, concluding that plaintiffs had standing, but that their challenge should be rejected on substantive Establishment Clause grounds. I have no firm opinion on the standing question, but the concurrence’s substantive analysis seems correct to me, especially given Marsh v. Chambers, on which the concurrence heavily relies.
Before oral argument, the plaintiffs had asked the court not to use the court’s standard invocation, “God save the United States and this honorable Court.” The concurrence also gives a justification for the denial of the motion, and that too strikes me as correct under current precedents.
UPDATE: I initially forgot to include a link to the opinion; just added it.