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Integrate[1/(a^2*Cos[x]^2 + b^2*Sin[x]^2), x] ==
ArcTan[(b*Tan[x])/a]/(a*b)

                     1
Integrate[-----------------------, x] == 
           2       2    2       2
          a  Cos[x]  + b  Sin[x]

       b Tan[x]
ArcTan[--------]
          a
----------------
      a b