Evaluate: `int_0^(pi//2)1/(cos^3xsqrt(2sin2x))dx`

Explanation

Take `I=int_{0}^{pi / 4} frac{d x}{cos ^{3} x sqrt{2} sin 2 x} d x`
`
=int_{0}^{pi / 4} frac{d x}{cos ^{3} x sqrt{2.2 sin x cdot cos x}}`
`
=frac{1}{2} int_{0}^{pi / 4} frac{{dx}}{cos ^{3} x sqrt{frac{sin x}{cos x} cdot cos ^{2} x}}`
`
=frac{1}{2} int_{0}^{x / 4} frac{d x}{cos ^{4} x sqrt{tan x}}`
`
=frac{1}{2} int_{0}^{pi / 4} frac{sec ^{4} x d x}{sqrt{tan x}}`
`
=frac{1}{2} int_{0}^{pi / 4} frac{sec ^{2} x cdot sec ^{2} x d x}{sqrt{tan x}}`

Take `tan x=t, x=0`
`
Rightarrow t=0 and x=frac{pi}{4}`
`
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