integration - A sine integral

The integral

$$\begin{align} \int_{0}^{\pi/2} \frac{ \sin(n\theta) }{ \sin(\theta) } \ d\theta \end{align}$$
is claimed to not have a closed form expression. In this view find the series solution of the integral as a series involving of $n$.

Editorial note:
As described in the problem several series may be obtained, of which, all seem to hold validity. As a particular case, from notes that were made a long while ago, the formula

$$\begin{align} \int_{0}^{\pi/2} \frac{ \sin(n\theta) }{ \sin(\theta) } \ d\theta = \sum_{r=1}^{\infty} (-1)^{r-1} \ \ln\left(\frac{2r+1}{2r-1}\right) \ \sin(r n \pi) \end{align}$$
is stated, but left unproved. Can this formula be proven along with finding other series dependent upon $n$?