Advice on method to integrate Bessel functions in C++

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I need to integrate an oscillatory function of this kind: enter image description here

where I have a Bessel function which is oscillating, while F is not very oscillatory. I am looking for the most precise/accurate method to do this in C++. Hopefully this should be an already existing implementation which I can use through for example a library etc...

The GSL library could be an option but please even in that case could you recommend me which of the many routines avaialble is the most useful for me?

EDIT: I am aware of the presence of this possibly duplicate question

but I don't see a clear answer there. Also, a Fortran library would not be good for me, unless there is some kind of wrapper for it.

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