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

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

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

                        x
            (a - b) Tan[-]
                        2
2 b ArcTanh[--------------]
                   2    2
            Sqrt[-a  + b ]            a Sin[x]
--------------------------- + ------------------------
   2    2         2    2        2    2
 (a  - b ) Sqrt[-a  + b ]     (a  - b ) (a + b Cos[x])
Time to compute: 0.28 second
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