Laplace of the square root of trig functions?

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I want to find the laplace of 1/rootcos(t)

Laplace calculators don't give an answer with this as an input. I know nothing about laplace, so can someone explain why this happens? Is it impossible to perform laplace on square roots? I'm assuming there's nothing wrong with the calculators.

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In case you want a nice formula (given by a CAS, be sure !) $$\mathcal{L}_t\left[\frac{1}{\sqrt{\cos(t)}}\right](s)=\Gamma \left(\frac{2i s+1}{4}\right) \left(i \, _2\tilde{F}_1\left(1,\frac{2 i s+3}{4} ;\frac{2 i s+3}{4} ;-1\right)+\frac{\left(1-i\right) \sqrt{\pi } e^{\frac{3 \pi s}{2}} (\coth (\pi s)-1)}{2\,\Gamma \left(\frac{2i s+3}{4}\right)}\right)$$