Why can't Mathematica do an easy integral?

Question

Mathematica can't do this integral: $\int \sqrt{1-x^2} \cos ^{-1}(x) \cos ^{-1}\left(\sqrt{1-x^2}\right) dx$ However if I replace $\cos ^{-1}\left(\sqrt{1-x^2}\right)$ with $\frac{\pi }{2}-\sin ^{-1}(x)$, it can handle.

What would increase the probability for Mathematica to be able to handle an integral?

Answer 1

I am still not sure if this is exactly what you want for your question, but the easiest way (in terms of least typing for me!) to evaluate this particular integral is probably with TrigToExp:

int = Sqrt[1 - x^2] ArcCos[x] ArcCos[Sqrt[1 - x^2]];
expRepresentation = int // TrigToExp // FullSimplify;

integral = Integrate[expRepresentation, x];

(D[integral, x] == int) // FullSimplify
(*True*)


integral // FullSimplify
(*(1/(24 x))Sqrt[x^2] (3 (-1 + 2 x^2) (-ArcCos[x] + ArcSin[x]) - 
   2 ArcCos[x]^2 (ArcCos[x] + 3 ArcSin[x]) + 
   6 x Sqrt[1 - x^2] (1 + 2 ArcCos[x] ArcSin[x]))*)