Answer by Ulrich Neumann for Area of ​a parametrically defined function

In addition to @cvgmt (see comment) here a vectorial solution v[\[Gamma]]using Cross

With[{v = {\[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +     Bmax Sin[\[Gamma]] \[Chi], (Bmax^2 - (Bmax Sin[             ArcTan[Hr/               Sqrt[He^2 + ((\[Alpha] Sinh[                    Bmax Sin[\[Gamma]] \[Beta]] +                    Bmax Sin[\[Gamma]] \[Chi]))^2]]])^2)^(1/         2) Sin[\[Gamma]] - (Bmax Sin[         ArcTan[Hr/           Sqrt[He^2 + ((\[Alpha] Sinh[Bmax Sin[\[Gamma]] \[Beta]] +                 Bmax Sin[\[Gamma]] \[Chi]))^2]]]) Cos[\[Gamma]]}},  NIntegrate[Cross[v] . D[v, \[Gamma]]/2, {\[Gamma], -\[Pi], \[Pi]}]] (* 558.924 *)