\[\large \int_0^{\pi/4}\dfrac{\sec x}{1+2\sin^2 x}\,dx\]

If the value of the above integral can be written as \(\dfrac{\sqrt A}B\pi^C + \dfrac1 B\ln\left(D+B\sqrt A\right)\), where \(A,B,C\) and \(D\) are positive integers with minimised \(A\), find \(A+2B+C+D\).

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