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

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

The following integral \(I\) is equal to \(\dfrac{\pi\sqrt{\alpha}+\beta\sqrt{\gamma}}{\delta},\) for positive square-free integers \(\alpha\) and \(\gamma\) and positive integers \(\beta\) and \(\delta.\) What is \(\alpha+\beta+\gamma+\delta?\)

\[I=\int_0^N\dfrac{\sec^4\theta}{\sec^4\theta-\tan^4\theta}\text{ }d\theta\]

\(\textbf{Details and Assumptions}\)

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