Sorry, I'm very confused. You say indefinite integral but then say it has limits - is that just a typo? The inverse of which function? Do you need to find the inverse of f(x) and then integrate with those limits or are you trying to find the indefinite integral and then find the inverse of that function? Or are you just trying to find the definite integral, and then do 1/result?
This seems to contradict what you were saying in the first quote.
Are you just trying to integrate ((tan(c) tan^(-1)(x tan(C)))/(x^2 tan^2(c) + 1) - (tan(C) tan^(-1)(x tan(c)))/(x^2 tan^2(C) + 1))/tan^(-1)(x tan(C))^2 ?
Is that result from the mean value theory? Doesn't make sense to me, you still have x in the result. Also doesn't make sense if that's not from MVT and it's from taking the derivative directly - for one, doesn't look like the correct derivative, and secondly, in that case integrating wouldn't accomplish anything - it would just return the original function and you seem to want to integrate the result.
Wouldn't you just set the MVT result equal to the derivative of f(x), and solve for x? In which case, what and why are you integrating? Is the final goal to get f(c) once you get c? As in, just plug c in for x in h(x) from the geogebra page?
Would like to try to help with the calculations and I've put most of your equations into mathcad, but I'm totally lost as to what you're trying to do.