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Integrate[1/Sqrt[a + b*Tan[c*x]^2], x] ==
((I/2)* (-Log[((-4*I)*(a - I*b*Tan[c*x] + Sqrt[a - b]* Sqrt[a + b*Tan[c*x]^2]))/(Sqrt[a - b]* (I + Tan[c*x]))] + Log[((4*I)*(a + I*b*Tan[c*x] + Sqrt[a - b]* Sqrt[a + b*Tan[c*x]^2]))/(Sqrt[a - b]* (-I + Tan[c*x]))]))/(Sqrt[a - b]*c)



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

 I
(- (-Log[((-4 I) (a - I b Tan[c x] + 
 2
 
                                            2
             Sqrt[a - b] Sqrt[a + b Tan[c x] ])) / 
 
         (Sqrt[a - b] (I + Tan[c x]))] + 
 
      Log[((4 I) (a + I b Tan[c x] + 
 
                                           2
            Sqrt[a - b] Sqrt[a + b Tan[c x] ])) / 
 
        (Sqrt[a - b] (-I + Tan[c x]))])) / 
 
  (Sqrt[a - b] c)

Time to compute: 0.95 second

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