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Integrate[x/Sin[a*x]^n, x] ==
(Sin[a*x]^(1 - n)* ((2*a*x*Cos[a*x]*Hypergeometric2F1[1, 1 - n/2, (3 - n)/2, Sin[a*x]^2])/(1 - n) - 2^(-1 + n)*Sqrt[Pi]*Gamma[1 - n]* HypergeometricPFQRegularized[{1, 1 - n/2, 1 - n/2}, {(3 - n)/2, 2 - n/2}, Sin[a*x]^2]* Sin[a*x]))/(2*a^2)



              x
Integrate[---------, x] == 
                  n
          Sin[a x]

         1 - n
(Sin[a x]      ((2 a x Cos[a x] 
 
                                  n  3 - n          2
         Hypergeometric2F1[1, 1 - -, -----, Sin[a x] ]
                                  2    2
 
         ) / (1 - n) - 
 
       -1 + n
      2       Sqrt[Pi] Gamma[1 - n] 
 
                                            n
       HypergeometricPFQRegularized[{1, 1 - -, 
                                            2
 
             n    3 - n      n           2
         1 - -}, {-----, 2 - -}, Sin[a x] ] Sin[a x]))
             2      2        2
 
          2
    / (2 a )

Time to compute: 0.36 second

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