fx = (1+x^2)^(-k) Exp[-a ArcTan[x]]

gx[k_,a_] := (1+x^2)^(-k) Exp[-a ArcTan[x]]

Plot[ gx[2,1], {x,-Pi,Pi} ]
Plot[ gx[4,1], {x,-Pi/2,Pi/2} ]
Plot[ gx[8,1], {x,-Pi/3,Pi/3} ]

val = D[ Log[ fx ], x ]
Simplify[ val ]