Interesting Integrals 7

Calculus Level 4

dxcos3x2sin(2x)=(tanx)A+C(tanx)B+k\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 AA, BB and CC such that A+B+C=pqA+B+C = \dfrac pq, where pp and qq are coprime positive integers and kk is the arbitrary constant of integration. Find p+qp+q.

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