Integral Calculus: JEE Main 2026 Mathematics question

If $\int(\sin x)^{-\frac{11}{2}}(\cos x)^{-\frac{5}{2}}\,dx=-\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}}+C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i)=1$ for $i=1, 2, 3, 4$ and C is the constant of integration, then $\frac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to