calculus and analysis – Mathematica returns incorrect evaluation of integral

The issue appears to be that Mathematica is handling the assumptions concerning x incorrectly. If you explicitly tell it that $x < -1$, $|x| leq 1$, or $x > 1$, it produces three slightly different functional forms:

ulow(x_) = Assuming(x < -1 && (Eta) > 0, ... )
umid(x_) = Assuming(1 > x > -1 && (Eta) > 0, ... )
uhigh(x_) = Assuming(1 < x && (Eta) > 0, ... )

(* (4 I ((Pi) + ArcSin(x)))/(x Sqrt(-1 + x^2)) *)
(* -((4 ArcCos(Sqrt(1 - x^2)))/Sqrt(x^2 - x^4)) *)
(* (4 I ArcSin(x))/(x Sqrt(-1 + x^2)) *)

So if desired, we can assemble these into a Piecewise function that works for all x:

u(x_) := Piecewise({{ulow(x), x< -1}, {umid(x), -1 <= x <= 1}, {uhigh(x), x > 1}}) 
Plot(Evaluate(ReIm(u(x))), {x, -Pi, Pi})

enter image description here

This does appear to have the property that $Re(u(x))$ is even in $x$ and $Im(u(x))$ is odd in $x$.