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(Log[-Cos[(a*x)/2] - Sin[(a*x)/2]] - Log[-Cos[(a*x)/2] + Sin[(a*x)/2]] + (Cos[(a*x)/2] + Sin[(a*x)/2])^(-2))/(2*a)


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



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

          a x        a x
(Log[-Cos[---] - Sin[---]] - 
           2          2
 
             a x        a x
    Log[-Cos[---] + Sin[---]] + 
              2          2
 
         a x        a x  -2
    (Cos[---] + Sin[---])  ) / (2 a)
          2          2

Time to compute: 0.11 second

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