This is probably an overkill way to a solution, but applying Euler reflection formula $$ \sin(z)=\frac{\pi}{\Gamma(1-\frac{z}{\pi})\Gamma(\frac{z}{\pi})}, $$ one gets products of gamma functions both in the numerator and denominator. From this point on, I think, your integral is computable/representable via a generalized Hypergeometric Function of Fox type --> http://en.wikipedia.org/wiki/Fox_H-function (that is Barnes-Integral with a kernel of fraction of products of Gamma functions). If my memory serves me right, this type of integral has been treated also in the book of Bleistein and Handelsman.
M.G.
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