Antiderivatives may differ by a constant over connected components of the domain. There is not just one antiderivative that is "right." Trying plotting the difference of the two expressions. It should consist of horizontal lines, perhaps more than one since Sqrt[expr] may have branch cuts. Sometimes one may need to plot something of the form ReImPlot[ (* difference *), {x, a, b}] or ComplexPlot3D[(* difference *), {x, a1 + a2*I, b1 + b2*I}], since Integrate[] really deals with antiderivatives over the complex plane where the branch cuts live.

Also consider that trig identities mean some functions may be written in terms of other functions.

FullSimplify[ArcTan[x/Sqrt[1 - x^2]] == ArcSin[x], -1 < x < 1]
(*  True  *)

So ArcTan[] may be expressed in terms of ArcSin[] and vice versa.