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-(Log[1 + b + (-1 + b)*Cos[2*a*x]]/(-2*a + 2*a*b))

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

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

  Log[1 + b + (-1 + b) Cos[2 a x]]
-(--------------------------------)
            -2 a + 2 a b
Time to compute: 0.09 second
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