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To get certified precision, use Arb via ComplexBallField.

Here is a way to use it to compute the integral in the question.

Define a complex ball field with, for instance, 100 bits or 200 bits of precision (which amounts to about 30 or 60 decimal digits).

Then compute the integral.

The answer is a ball given by its center and a radius around it.

sage: C = ComplexBallField(100)
sage: C.integral(lambda x, _: 2*arcsin(x/(sqrt(1-x^2))), 0, ~AA(3).sqrt())
[0.3833009065188100522199318983 +/- 6.39e-29]

sage: C = ComplexBallField(200)
sage: C.integral(lambda x, _: 2*arcsin(x/(sqrt(1-x^2))), 0, ~AA(3).sqrt())
[0.3833009065188100522199318982744940521091715574838526509558 +/- 5.35e-59]

By contrast, numerical_integral gives an error estimate which, if I understand correctly, is not certified.