I am trying to find a counterexample to the LIATE rule to demonstrate to students.
The ones I've found ($x^3e^{x^2}, \frac{x\sin x}{1+x^2}$, etc) are all examples where part of the integrand is a substitution integration in disguise and integrating by parts following the LIATE order destroys that substitution. I personally find this less satisfying.
I'm wondering if there are any examples where the integral is a straightforward integration by parts but is difficult or impossible using the LIATE order.
Here's an interesting example I came across:
$\displaystyle \int \dfrac{xe^{2x}}{(1+2x)^2}dx$
If you choose $u = \dfrac{x}{(1+2x)^2}$ (which is what LIATE would choose) for the integration by parts, things are going to get messy, but if you choose $u = xe^{2x}$, then it simplifies quite nicely.