Wolfram's
online integrator could not come up with an answer when differing numbers were plugged in for "a" and "b". This suggests that your conclusion may be correct....
Note: I first used a = 1, b = 2, and entered "x/Sqrt[(Exp[x]-1)(2-Exp[x])]" in the box. When I plugged "2" in for each, The Integrator returned the following:
. . . . ....where "Li
_{2}" is something called the "polylog function"...?