X[θ_] := {4 Cos[θ] + Cos[4 θ], 
        4 Sin[θ] + Sin[4 θ]}
X[θ]
ParametricPlot[X[θ], {θ, 0, 2 Pi},  
      AspectRatio -> Automatic]

v[θ_] := (∂_θ) X[θ]

v[θ]
Expand[v[θ] . v[θ]]
speed[θ_] := (Sqrt[Simplify])[Expand[v[θ] . v[θ]]]

(∫_0^(2 π))speed[θ] \[DifferentialD]θ

s[t_] :=  
      NIntegrate[speed[θ], { θ,  0,  t}]

Off[NIntegrate::ncvb]
Off[NIntegrate::slwcon]

Plot[s[t], { t,  0,  2  Pi}]