If
\[
\int \left( \cos x \right)^{-\frac{5}{2}} \left( \sin x \right)^{-\frac{1}{2}} \, dx
\]
\[
= \frac{P_1}{q_1} \left( \cot x \right)^{\frac{3}{2}} + \frac{P_2}{q_2} \left( \cot x \right)^{\frac{3}{2}} + \frac{P_3}{q_3} \left( \cot x \right)^{\frac{1}{2}} + \frac{P_4}{q_4} \left( \cot x \right)^{-\frac{3}{2}} + c
\]
(where \( c \) is the constant of integration), then value of
\[
\frac{15P_1P_2P_3P_4}{q_1q_2q_3q_4}
\]
is: