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Integrate[E^(a*x)/Sin[x]^n, x] ==
(E^(a*x)*(2 - 2*E^((2*I)*x))^n*Hypergeometric2F1[ (-I/2)*(a + I*n), n, (2 - I*a + n)/2, E^((2*I)*x)])/((((-I)*(-1 + E^((2*I)*x)))/E^(I*x))^ n*(a + I*n))

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

  a x         (2 I) x n
(E    (2 - 2 E       )  
 
                      -I               2 - I a + n
    Hypergeometric2F1[-- (a + I n), n, -----------, 
                      2                     2
 
                              (2 I) x
      (2 I) x       -I (-1 + E       ) n
     E       ]) / ((------------------)  (a + I n))
                            I x
                           E