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∫
(Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, Tanh[x]^2]*Tanh[x]^(1 + n))/(1 + n)

Integrate[Tanh[x]^n, x] ==
(Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, Tanh[x]^2]*Tanh[x]^(1 + n))/(1 + n) 

                 n
Integrate[Tanh[x] , x] == 

                      1 + n      1 + n         2
(Hypergeometric2F1[1, -----, 1 + -----, Tanh[x] ] 
                        2          2
 
           1 + n
    Tanh[x]     ) / (1 + n)
Time to compute: 0.07 second
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