There are many cases when product rule of integration proves to be cumbersome and it may not work. While doing an integral: $$\int \frac{\log(t)}{1+t}dt$$
I found that the product rule fails, though it apparently seems to be applicable. Are there any limitations to this rule? Also, please suggest an alternate way of solving the above integral.
It's called Integration by Parts, and it doesn't "fail". It may happen, though, as it occurs in the example you mention, that it doesn't lead to anything useful (at least not in a direct way).
As far as I can tell, the antiderivative in your question doesn't have an expression in terms of elementary functions. So it is not about tricks.