Can *Mathematica* solve integral $(1)$?

$$\int \frac{\cot^2\left(\frac12\sec^{-1}(x)\right)}{\sqrt{\tan\left(\frac12\csc^{-1}(x)\right)}} \, dx.\tag{1}$$

This is 

`Integrate[Cot[1/2 ArcSec[x]]^2 / Sqrt[Tan[1/2 ArcCsc[x]]], x]`

Perhaps it is worth clarifying that it is not that I do not know how to solve it. The integral can be mechanically solved using the method described in this [blog post][1]. But I have noticed that both [Wolfram Alpha][2] and this [Integral Calculator][3] are unable to simplify it, let alone solve it (I mean they do not provide the closed form of the integral).


  [1]: https://geometriadominicana.blogspot.com/2024/03/integration-using-some-euler-like.html
  [2]: https://www.wolframalpha.com/input?i=integrate%20cot%5E2%281%2F2arcsec%28x%29%29%2Fsqrt%28tan%281%2F2arccsc%28x%29%29%29
  [3]: https://www.integral-calculator.com/#expr=cot%5E2%281%2F2arcsec%28x%29%29%2Fsqrt%28tan%281%2F2arccsc%28x%29%29