I understand how they obtained the inversion of sin(x) shown here, using the Lagrange Inversion Formula, and have even written a MATLAB script to solve the inversion when input and output exponents are integers.
However, I don't see how they computed the inversion of x*sin(x) (first link). The result uses fractional powers, which I assume relates to 1/GCD{input exponents}.
How do you solve for the inversion when either the input or output exponents can be fractional? Is there a worked out example somewhere I can use as reference? At first thought I could factor out the common x^m and go from there, but it doesn't seem to be correct method. I would like to eventually be able to generalize the code to handle those cases - but even understanding the theory behind it would be great in of itself.