DavidHooper: I am trying to integrate the reciprocal of the function you posted. Also, I think you have a typo in your result. It looks as though you have an extra right parenthesis, which I assume is a divison operator. If this assumption is correct, the result you specified is the following.

SquareRoot( X^2 + [X - cos(A*R)]^2 ) / A, whose derivative seems to me to be the following.

[X - cos(A*R)] * sin(AR) / SquareRoot( X^2 + [X - cos(A*R)]^2 )

The above derivative is not equal to either your integrand or my integrand.