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



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

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

Time to compute: 0.18 second

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