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Integrate[x/(b + Tan[a*x]), x] ==
(x^2 - (a^2*E^(I*ArcTan[b])*x^2 + (b*((-I)*a*x*(Pi - 2*ArcTan[b]) - Pi*Log[1 + E^((-2*I)*a*x)] - 2*(a*x + ArcTan[b])* Log[1 - E^((2*I)*(a*x + ArcTan[b]))] + Pi*Log[Cos[a*x]] + 2*ArcTan[b]* Log[Sin[a*x + ArcTan[b]]] + I*PolyLog[2, E^((2*I)*(a*x + ArcTan[b]))]))/ Sqrt[1 + b^2])/(a^2*Sqrt[1 + b^2]))/(2*b)



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

  2     2  I ArcTan[b]  2
(x  - (a  E            x  + 
 
       (b (-I a x (Pi - 2 ArcTan[b]) - 
 
                        (-2 I) a x
            Pi Log[1 + E          ] - 
 
            2 (a x + ArcTan[b]) 
 
                      (2 I) (a x + ArcTan[b])
             Log[1 - E                       ] + 
 
            Pi Log[Cos[a x]] + 
 
            2 ArcTan[b] Log[Sin[a x + ArcTan[b]]] + 
 
                          (2 I) (a x + ArcTan[b])
            I PolyLog[2, E                       ]))\
 
                     2       2           2
         / Sqrt[1 + b ]) / (a  Sqrt[1 + b ])) / (2 b)

Time to compute: 1.05 second

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