Deriving Integral of cos(tan(θ)-2θ) from 0 to pi/2

[zack_vt]

A certain program, that shall remain nameless, tells me that the integral of cos(tan(θ)-2θ) from 0 to pi/2 is pi/e. Does anyone know of a variable transform or contour that would help me to derive this result? I'm really interested in integrals of the function cos(a tan(θ)-b θ), but, I'd be temporarily contented with knowing how to explicitly do the (a=1,b=2) case.

 

[JJacquelin]

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