First of all, let me clarify that I think that this question is on-topic here, as symbolic integration is a problem that also humans solve, so it requires some kind of intelligence.
Second, I had also watched that interesting lecture by Winston a few months ago, so I remember that some of the rules that he mentions during the lecture are just rules that humans sometimes also use to solve integrals (e.g. integration by parts, which you should have an idea of, if you ever took a calculus course in high school), but he also mentions other problem-solving techniques used in AI, like And–Or trees.
That being said, I've never implemented any symbolic integration program, but I think that the paper A heuristic program that solves symbolic integration problems in freshman calculus (1963), by James R. Slagle, could (at least partially) answer this question, as it describes a symbolic integration program, SAINT (which stands for Symbolic Automatic INTegrator), and it also mentions a few transformations/rules that it uses, and the And-Or goal trees, which are also mentioned by Winston in that lecture, so I guess Winston may have been referring to the techniques in this or a similar paper (I would need to rewatch the lecture to confirm this). More details about SAINT are given in Slage's 1961 Ph.D. dissertation.
I suspect that, nowadays, there may be more efficient programs than SAINT to solve symbolic integration, so I will leave the details to an expert on the topic.
In addition to that, you can find on the web articles (for example, this, this, and this) that describe the common integration rules.