I have yet to encounter a philosophical argument against libertarian free will that does not commit a straw man fallacy. The error usually occurs when an opponent of libertarian free will tries to give some sort of exhaustive definition of libertarian free will. After giving the definition, they will then attack it for being unintelligible or incoherent. This very line of reasoning is fallacious. Here's why:
Some terms can be defined exhaustively. A circle, for example, can be defined exhaustively as a closed plane curve consisting of all points at a given distance from a point within it called the center. There is nothing more to a circle than that. Other terms, such as time, liberty, knowledge, and cause, have resisted all attempts for philosophers to give an exhaustive definition. Sure, they can be defined, but we generally do not know them through their definition. We can pick out instances of these things, even if we cannot give a complete definition of such terms. Free will is like that. We acknowledge it every time we give a person praise or blame for their actions, and every time we hold someone morally accountable, such as in a trial. Hence, these terms can only be given loose or general (non-exhaustive) definitions. Here are such definitions for free will and libertarian free will.
Free will: the unique ability of persons to exercise control over their conduct in the fullest manner necessary for moral responsibility.
Libertarian Free Will: Non-deterministic free will.
Any other definition for either of these terms is a straw man fallacy. Once we acknowledge these definitions, however, the entire philosophical case against the coherence of free will disappears.