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This integral appears on solving a particular problem of gamma ray detectors. I have tried several substitutions and methods but can't solve it. Maybe it uses some special functions.

$$\int_{\theta_1}^{\theta_2} \cos^{-1}(a\hspace{0.05cm}\cot \theta)e^{-b\hspace{0.05cm}\sec(\theta)}\sin(\theta)d\theta$$

where $a=\frac{l}{y}cos(\frac{\pi}{n})$ and b are constants.

$tan(\theta_1)=a$ and $tan(\theta_2)=asec(\frac{\pi}{n})$

How to solve it?

Asit Srivastava
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    Put this into an integrator. Like Wolfram Alpha. – Allawonder Aug 21 '19 at 22:51
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    I would substitute $u = \cos x$, then $v=1/u$ and then use integration by parts or Feynman's trick to turn $\arccos$ into an algebraic function. Then it might lead to some Bessel functions, though indefinite integral doesn't look promising – Yuriy S Aug 21 '19 at 22:53
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    Anything better than an infinite series is unlikely – Yuriy S Aug 21 '19 at 22:56
  • There was another integral also that reduced to exponential integral function. Although it can't be solved yet it was in a simple form and could be looked into tables directly for limits. Is it possible to reduce this integral to some simple form even though not integrable? – Asit Srivastava Aug 21 '19 at 23:07
  • I couldn't find the limits earlier but have found them now. Sorry for not adding them before. – Asit Srivastava Aug 21 '19 at 23:45

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