SolveByTakingASignOut​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

The sign of a real-valued function of x among the factors of the
integrand is constant on every interval between its zeros, and goes in front of the
antiderivative of the rest: sgn(x) f(x) is sgn(x) F(x) on each half-line,
which is the generic case every rule answers in. The extension by parity writes the
antiderivative of x^2/sqrt(1/(a x)^2 - 1) with a sgn(x) in front, and a
step of parts against it -- x^2 asech(a x)/sqrt(1/(a x)^2 - 1), the second step
on x^3 asech(a x)^2 -- leaves that sign beside the remainder, where no rule
read past it. Only a sign whose argument is shown real: for a complex g,
sgn(g) is g/|g| and is not constant anywhere. Asked as the same question.
https://github.com/asc-community/AngouriMath/issues/718