From tangent to cotangent

Calculus Level 3

\[ \large \int_{\tan x}^{\cot x} \dfrac{\ln(t^{2n})}{1+t^2} \, dt \]

Let \(x \in \left( 0 ,\dfrac\pi2\right) \), and \(n\) be any positive integer. Compute the integral above.

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