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(2*b*ArcTan[Tan[a*x]] - Log[Sec[a*x]^2] + 2*Log[b + Tan[a*x]])/(2*a + 2*a*b^2)

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

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

                                    2
(2 b ArcTan[Tan[a x]] - Log[Sec[a x] ] + 
 
                                       2
    2 Log[b + Tan[a x]]) / (2 a + 2 a b )
Time to compute: 0.08 second
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