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-((Cot[a*x]^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Cot[a*x]^2])/(a + a*n))


Integrate[Cot[a*x]^n, x] ==
-((Cot[a*x]^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Cot[a*x]^2])/(a + a*n))



                  n
Integrate[Cot[a x] , x] == 

           1 + n                      1 + n  3 + n
-((Cot[a x]      Hypergeometric2F1[1, -----, -----, 
                                        2      2
 
                2
       -Cot[a x] ]) / (a + a n))

Time to compute: 0.09 second

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