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ArcTan[(Sqrt[a^2 + b^2]*Tan[c*x])/a]/(a*Sqrt[a^2 + b^2]*c)

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

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

             2    2
       Sqrt[a  + b ] Tan[c x]
ArcTan[----------------------]
                 a
------------------------------
              2    2
      a Sqrt[a  + b ] c
Time to compute: 0.12 second
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