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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?

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    $\begingroup$ IntegrateChangeVariables works with substitution $t = \sqrt{1-x^2}$ but I am not sure if this is what you are asking. I think you are asking for a way to make Integrate evaluate this integral without any sort of user intervention by using another function, right? $\endgroup$
    – ydd
    Commented Sep 6, 2023 at 0:04
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    $\begingroup$ People here generally like users to post code as copyable Mathematica code as well as images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find the meta Q&A, How to copy code from Mathematica so it looks good on this site, helpful $\endgroup$
    – Michael E2
    Commented Sep 6, 2023 at 0:35
  • $\begingroup$ I suspect it gets lost in the branch cuts. $\endgroup$
    – John Doty
    Commented Sep 6, 2023 at 13:53

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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]))*)
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