Interesting Integrals 7

Calculus Level 4

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

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 $p+q$ .