You've probably seen this integral before if you paid attention

Calculus Level 3

NN is the smallest angle, in radians, of a triangle with side lengths 4,4, 5,5, and 41810.\sqrt{41-8\sqrt{10}}.

The following integral II is equal to πα+βγδ,\dfrac{\pi\sqrt{\alpha}+\beta\sqrt{\gamma}}{\delta}, for positive square-free integers α\alpha and γ\gamma and positive integers β\beta and δ.\delta. What is α+β+γ+δ?\alpha+\beta+\gamma+\delta?

I=0Nsec4θsec4θtan4θ dθI=\int_0^N\dfrac{\sec^4\theta}{\sec^4\theta-\tan^4\theta}\text{ }d\theta


Details and Assumptions\textbf{Details and Assumptions}

There is exactly one\textbf{one} place in this problem where you may need to use a four-function calculator.

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