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

An exponential of i times an inverse trigonometric function, written
algebraically: e^(i arctan(L)) is (1 + i L)/sqrt(1 + L^2),
e^(i arcsin(L)) is sqrt(1 - L^2) + i L and e^(i arccos(L)) is
L + i sqrt(1 - L^2), each on the principal branch for a real L, and
e^(n i f(L)) is that to the nth, for a rational n. Rubi's
e^(i arctan(a x))/sqrt(c + a^2 c x^2) is then (1 + i a x)/(sqrt(c) (1 + a^2 x^2)),
a rational function; under x = tan(u)/a it was e^(i u) sec(u), which
nothing in u reads. The integrand as rewritten is asked as a question of
its own, and the answer checked at sampled points where a symbol is involved.