Question: looking for a continous anti-derivate for trig/rational

I want a continous anti-derivate (cos(2*u*m)-alpha*u*sin(2*u*m))/(4*u^2+1)
w.r.t. u, for alpha and m Reals, 0 < u.

What I get with Maple in terms of Ci and Si jumps in m=0 (and for m=0 it
is not defined, due to Ci).

The best I got was

  Si(-conjugate(v))-Si(v); eval(%, v=m*(2*u+I));
  psi:= unapply(%, u,m);
  chi:= (u,m) -> piecewise(0 <= m, -Ci((2*u-I)*m)+Ci(m*(2*u+I)), 
                            m < 0, -Ci((2*u-I)*m)+Ci(m*(2*u+I)) + 2*I*Pi);


as one can see from examples like

  subs(u=1/2,alpha=2, L);
  plot([Re(%), Im(%)], m=-1..1);

To see that L solves the task one differentiates it w.r.t. and uses

  L assuming (0 <m); # or the converse
  diff(%,u); convert(%, expln): evalc(%): collect(%,exp): combine(%,exp): sort(%)

Also one can add a constant (w.r.t. u) to chi.

Any nicer idea for that? My special concern is Numerics to cover m ~ 0, for
which I am not sure to switch to Ci = Sum(...) + ln() + gamma [ according to
the FunctionAdvisor for Ci, ln can be treated similar by piecewise].
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