Interesting Integrals 7

Calculus Level 5

\[\large \int\dfrac{dx}{{\cos^3 x}\sqrt{2\sin(2x)}}\,= {(\tan x)^A} + C{(\tan x)^B}+k \]

The equation above holds true for reals \(A\), \(B\) and \(C\) such that \(A+B+C = \dfrac pq\), where \(p\) and \(q\) are coprime positive integers and \(k\) is the arbitrary constant of integration. Find \(p+q\).