Why is Mathematica unable to solve an integral than can solved with WolframAlpha? The integral in question is this: $$\frac{1}{2}\int_0^{2\pi} [\cos^{-1}(\sin x)]^2~dx$$ The Mathematica code is **Integrate[(Acos[Sin[x]])^2/2, {x, 0, 2 Pi}]** Attempting this with Mathematica appears to be fruitless. It won't even do a numerical integration. WolframAlpha, however, deftly provides $$\frac{1}{2}\int_0^{2\pi} [\cos^{-1}(\sin x)]^2~dx=\frac{\pi^3}{3}\approx 10.335$$ However, WA also declares a step-by-step solution is unavailable. Incidentally, I know that the WA solution is correct because I have verified it numerically.

Well, because Mathematica is a programming language and has a defined syntax. The function is called `ArcCos` and not _Acos_ :


Integrate[(ArcCos[Sin[x]])^2/2, {x, 0, 2 Pi}]
(* Pi^3/3 *)