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

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

                        -2
Integrate[(a + b Cos[x])  , x] == 

                         x
             (a - b) Tan[-]
                         2
-2 a ArcTanh[--------------]
                    2    2
             Sqrt[-a  + b ]
---------------------------- + 
    2    2         2    2
  (a  - b ) Sqrt[-a  + b ]
 
          b Sin[x]
  -------------------------
     2    2
  (-a  + b ) (a + b Cos[x])
Time to compute: 0.22 second
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