In[]:=
Module[{a=RandomReal[{0,100}],b=RandomReal[{0,10}],c,A,A1,A2,B,α,β,γ,μ,ν},c=(a+)/b;A=(1+Abs[c]^2+Abs[c^2-1])/(2Re[c]);A1=A^2-2ARe[c]+Abs[c]^2;A2=A^2Abs[c]^2-2ARe[c]+1;B=A2/A1;α=cA-1;β=c-A;γ=2bSqrt[A1]Sqrt[A^2-1];μ=1/Sqrt[B+1];ν=1/Sqrt[A^2-1];{NIntegrate[Sqrt[1+(a-bSin[t])^2],{t,0,2Pi}],2γν^2/μ(EllipticE[μ^2]-EllipticK[μ^2]+(1+ν^2)EllipticPi[-ν^2,μ^2])}]
Out[]=
{481.2,481.2}