In[1]:= a = 2;
In[2]:= b[t_] = 3/10 + I + 5/2 Exp[I t];
In[3]:= z[t_] = 1/2 (b[t] + a^2/b[t])
Out[3]= 1/2 ((3/10 + I) + (5 E^(I t))/2 +
4/((3/10 + I) + (5 E^(I t))/2))
In[7]:= reimz[t_] = {Re[z[t]], Im[z[t]]} // ComplexExpand //
FullSimplify
Out[7]= {1/
20 (3 + 25 Cos[t]) (1 + 200/(367 + 75 Cos[t] + 250 Sin[t])), -(1/
2) + (5 Sin[t])/4 + (267 + 75 Cos[t])/(
367 + 75 Cos[t] + 250 Sin[t])}
In[12]:= dz[t_] = D[reimz[t], t] // FullSimplify
Out[12]= {-((
250 (3 + 25 Cos[t]) (10 Cos[t] - 3 Sin[t]))/(367 + 75 Cos[t] +
250 Sin[t])^2) -
5/4 Sin[t] (1 + 200/(367 + 75 Cos[t] + 250 Sin[t])),
5/4 (Cos[t] - (
600 (25 + 89 Cos[t] + 10 Sin[t]))/(367 + 75 Cos[t] +
250 Sin[t])^2)}
In[16]:= NIntegrate[Sqrt[dz[t].dz[t]], {t, 0, 2 \[Pi]}]
Out[16]= 9.40235
