Question: Why does this integral give undefined?

In the attached maple worksheet integral.mw, I am trying to compute an integral involving hyperbolic cosines. While not straightforward to compute, it should be completely well behaved at zero and both infinities as long as rho is real:

Much to my surprise, maple gives "undefined" as the answer, even if I attach an "assuming rho > 0" to make completely sure that the denominator never vanishes.

If I fix rho to 1 (or any other value) and compute the integral numerically, maple happily gives a numeric value. Moreover, I can scale the R, use the fact that the integrand is an even function of R, factor the denominator, and use partial fraction decomposition to transform it into the equivalent form

which maple integrates happily and gives a completely well-defined function in terms of polylogarithms, which agrees with the numeric answer.

Now, I am not expecting maple to just solve every integral I throw at it, but why does it give "undefined" as the answer to a completely well-defined integral?