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Why this simply elliptic integral fails


Hi,
This result seems to me to be wrong (it's right?):

In[1]:=
Integrate[Sqrt[a^2*Cos[t]^2 + b^2*Sin[t]^2],
  {t, 0, 2*Pi}, Assumptions->{Im[a]==0, Im[b]==0}]

Out[1]=
0

....but written in another way, it's right:

In[2]:=
Integrate[Sqrt[a^2 + (b^2 - a^2)*Sin[t]^2],
  {t, 0, 2*Pi}, Assumptions ->
   {Im[a] == 0, Im[b] == 0}]

Out[2]=
If[a^2/(-a^2 + b^2) >= 0 ||
   b^2/(-a^2 + b^2) <= 0 ||
   Im[a^2/(-a^2 + b^2)] != 0,
  (4*a^2*Sqrt[-1 + b^2/a^2]*EllipticE[
     1 - b^2/a^2])/Sqrt[-a^2 + b^2],
  Integrate[Sqrt[a^2 + (-a^2 + b^2)*Sin[t]^2],
   {t, 0, 2*Pi}]]

Someone can explane me this difference?
Thanks,
Raf.



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