Expressing an integral in closed form Is there a closed-form expression for this integral?

$$\int \frac{\sin(Ax/2)}{A\sin(x/2)}\mathrm{d}x$$
Mathematica gives
(after some editing)

$\int\frac{sin(ax)}{sin(x)}dx
=\dfrac{i*((1 + a)*Hypergeometric2F1[1, 1/2 - a/2, 3/2 - a/2, 
e^{(2*i)*x}] + (-1 + a)*e^{(2*i)*a*x}* Hypergeometric2F1[1, (1 + a)/2, (3 + a)/2, e^{(2*i)*x}])}{(-1 + a^2)*e^{i*(-1 + a)*x}}$
Mathematica gives
(after some editing)

$\int\frac{sin(ax)}{sin(x)}dx
=\dfrac{i*((1 + a)*Hypergeometric2F1[1, 1/2 - a/2, 3/2 - a/2, 
e^{(2*i)*x}] + (-1 + a)*e^{(2*i)*a*x}* Hypergeometric2F1[1, (1 + a)/2, (3 + a)/2, e^{(2*i)*x}])}{(-1 + a^2)*e^{i*(-1 + a)*x}}$