S := t -> tanh(ln(1 + t^2))

diff(S(t), t $ n)=1/GAMMA(1-n)-1/4*pochhammer(-n,n)*Sum(_alpha^3*(t-_alpha)^(-1-n),_alpha =RootOf(_Z^4+2*_Z^2+2))

why the root of? its easy to see the roots are given by

solve(_Z^4 + 2*_Z^2 + 2 = 0);

(-1-I)^(1/2), -(-1-I)^(1/2), (-1+I)^(1/2), -(-1+I)^(1/2)

its equivalent to the roots of (z^2 + 1)^2 + 1

it seems very weird that maple stops here rather than turning it into a 4-element array