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

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

            Csc[a x]
Integrate[------------, x] == 
          1 + Cos[a x]

          a x 2          a x             a x
1 - 2 Cos[---]  (Log[Cos[---]] - Log[Sin[---]])
           2              2               2
-----------------------------------------------
              2 a (1 + Cos[a x])
Time to compute: 0.07 second
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