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


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



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

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

Time to compute: 0.17 second

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