Integrate[(Sin[n t/2]/Sin[t/2])^2, {t, 0, Pi}]~Table~{n, 10} // Timing

{2.17188, {π, 2π, 3π, 4π, 5π, 6π, 7π, 8π, 9π, 10π}}

Integrate[(Sin[n t/2]/Sin[t/2])^2, {t, 0, Pi},
Assumptions -> n ∈ Integers] // Timing
%[[-1]] ~ FunctionExpand ~ (Assumptions -> n ∈ Integers)
% // FullSimplify

-(1/2) I E^(-I n π) (1 + E^(2 I n π) +
2 E^(I n π) n π Cot[n π] + n PolyGamma[0, 1/2 - n/2] -
n PolyGamma[0, -(n/2)] + E^(2 I n π) n PolyGamma[0, n/2] -
E^(2 I n π) n PolyGamma[0, (1 + n)/2])