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(-2*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2]

Integrate[1/(a + b*Cos[x]), x] ==
(-2*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/ Sqrt[-a^2 + b^2] 

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

                       x
           (a - b) Tan[-]
                       2
-2 ArcTanh[--------------]
                  2    2
           Sqrt[-a  + b ]
--------------------------
             2    2
      Sqrt[-a  + b ]
Time to compute: 0.06 second
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