Mathematics Question from JEE Main 2026 · 22 January, Shift 1

If $\int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} \, dx$ is equal to $-\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} -\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} -\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} +\frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C,$ where $p_i, q_i$ are positive integers with $\gcd(p_i,q_i)=1$ for $i=1,2,3,4$, then the value of $\frac{15\,p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is _____.