Some messy evaluation of integrals that end up expressed in terms of elliptic  E() and F() functions.  Was associated with the evaluation of the charge of a circular segment of line charge.

ClearAll[u, t, ii, f1, f2]
$Assumptions = u > 0 && u < 1 && t > 0 ;
f1[p_, u_,t_] := (1 + u^2 - 2 u Sin[t] Cos[p])^(3/2);
f2[p_, u_, t_] := E^(I p) f1[p, u, t];
ii1 = Integrate[ f1[p, u, t], p];
ii2 = Integrate[ f2[p, u, t], p];
ii1 //TraditionalForm
ii2 // TraditionalForm