x[theta_]  :=  {4 Cos[theta]  -  
          Cos[4 theta],  
4 Sin[theta]  -  
          Sin[4 theta]}

ParametricPlot[x[theta],  {theta,  0,  2 Pi},  
      AspectRatio  ->  1]

xprime[theta_]  =  x'[theta]

Clear[xprime]
xprime[theta_]  =  TrigExpand[x'[theta]]

Simplify[xprime[theta] . xprime[theta]]
speed[theta]  =  Sqrt[%]

(∫_0^(2 Pi))speed[theta] \[DifferentialD]theta