(* 4.382.5*)

Integrate[Log[((x+a)^2+c^2)/((x-a)^2+c^2)]*Sin[b*x],{x,0,Infinity},Assumptions->Re[c]>0&&Abs[Im[a]]<Re[c]&&(b>0)]

(* This is correct, but how do we simplify the answer given by Mathematica? *)

Integrate[Log[((x+a)^2+c^2)/((x-a)^2+c^2)]*Sin[b*x],{x,0,p},Assumptions->Re[c]>0&&Abs[Im[a]]<Re[c]&&(b>0)&&(p>0)]