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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[Sec[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)



            Sec[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: 0.13 second

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