I am sorry for putting multiple questions in the same post, but I think providing answers here will be better.
As far as I know, there is no 'product formula' for integrals, like we have for the derivative. Also, I can be wrong, but I think a general class of functions, differing by a constant, have the same derivative. So, ignoring the constant, one might think that such a product formula might exist. So, my first question:
Can it be proved that there is no 'product formula' for integrals, or is it just that it has not been discovered yet?
Let us reduce our case to just rational functions. Partial fractions for integration is a pretty good technique, but I think it can't be used for all rational functions, because not all polynomials have all real roots. So:
Is there a technique/algorithm to integrate all rational functions?
Another open-ended question, that I think of, is that are there any techniques/formulas for integrals not discovered yet, or are the existence of these techniques/formulas been proven false by some theorem? Answer to any question is suffice.
EDIT FROM HERE
After reading some of the answers, I felt I need to be more precise. The integration by parts formula, as far as I know, is again a heuristic, and not mechanical. So, there is no scope for integration by parts theorem to be such a product formula. Another user answered that I thought of a formula combining functions in an elementary way, and proposed that it is well known that the sinc function does not have a closed form integral, and so such a formula doesn't exist. But, to add to this, there is also a possibility that such a formula may produce an undefined result, or some weird result, which we can relate to the absence of such a closed-form solution. What I am looking for is a theorem or result which clearly proves the non-existence of such a formula, taking into account all possibilities.