Thanks to @OscarBenjamin, I learned that the SymPy function expand_trig() (but not method, which I tried) does the job: indeed,

from sympy import S,expand_trig
tanh,artanh = S("tanh, artanh")
def f(n):
    n=S(n)
    return expand_trig(tanh(sum(artanh(k/n) for k in range(1,n))))

f(3)  # does return 9/11 !