Answers.Body: <p>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.</p>
  <p>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. <a href="https://en.wikipedia.org/wiki/Integration_by_parts" rel="nofollow noreferrer"><strong>integration by parts</strong></a>, 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 <a href="https://en.wikipedia.org/wiki/And%E2%80%93or_tree" rel="nofollow noreferrer"><strong>And–Or trees</strong></a>.</p>
  <p>That being said, I've never implemented any symbolic integration program, but I think that the paper <a href="https://dl.acm.org/doi/10.1145/321186.321193" rel="nofollow noreferrer">A heuristic program that solves symbolic integration problems in freshman calculus</a> (1963), by James R. Slagle, could (at least partially) answer this question, as it describes a symbolic integration program, <strong>SAINT</strong> (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 <a href="https://dspace.mit.edu/bitstream/handle/1721.1/11997/31225400-MIT.pdf?sequence=2" rel="nofollow noreferrer">Slage's 1961 Ph.D. dissertation</a>.</p>
  <p>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.</p>
  <p>In addition to that, you can find on the web articles (for example, <a href="https://www.mathsisfun.com/calculus/integration-rules.html" rel="nofollow noreferrer">this</a>, <a href="https://math24.net/indefinite-integral-basic-rules-integration.html" rel="nofollow noreferrer">this</a>, and <a href="https://en.wikipedia.org/wiki/Lists_of_integrals" rel="nofollow noreferrer">this</a>) that describe the common integration rules.</p>
  
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Answers.Text: That is the answer to my question. I've looked at some of the websites, however, I was looking for the list. My flaw was not looking for a paper! Thanks.
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Answers.Text: Some interesting symbolic methods can also be found in the papers citing the Slagle's paper: [Citing papers](https://dl.acm.org/action/ajaxShowCitedBy?doi=10.1145/321186.321193)
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Body: <p>I want to implement a full symbolic integration. To achieve this. I've learned from Prof. <a href="https://youtu.be/PNKj529yY5c?t=2464" rel="nofollow noreferrer" title="Lecture on Youtube starts on the 41th minute">Patrick Winston's AI lecture that Matlab</a> uses</p>
  <ul>
  <li>12 safe transformations, like that constant out, sums, etc.</li>
  <li>12 heuristic transformation e.g. sin x, cos x...</li>
  <li>and, a table that contains 26 anchors to the calculations.</li>
  </ul>
  <p>So that every integration can be calculated using these rules.</p>
  <p>In the lecture, they only listed 4 of the safe transformations, 3 of the 12 heuristic transformations, and 3 of the 26 anchors.</p>
  <p>In order to implement the full symbolic integration, I need the rules. So; what are those? or where can I find them?</p>
  
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Comments.Text: If this does not suit this site, let me know.
Comments.UserDisplayName: kelalaka
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Comments.Text: Hello, welcome to AI stack exchange. Please provide us with more information about the problem. Update with the necessary links and provide more content for the experts to understand in detail.
Comments.UserDisplayName: hanugm
Tags: <reference-request><integration>
Title: What are the base rules for the symbolic integration implementation?
UserDisplayName: kelalaka
ThreadText: # What are the base rules for the symbolic integration implementation?
  
  (Asked by: username_0 on 2022-01-08)
  
  I want to implement a full symbolic integration. To achieve this. I've learned from Prof. [Patrick Winston's AI lecture that Matlab](https://youtu.be/PNKj529yY5c?t=2464 "Lecture on Youtube starts on the 41th minute") uses
  
  * 12 safe transformations, like that constant out, sums, etc.
  * 12 heuristic transformation e.g. sin x, cos x...
  * and, a table that contains 26 anchors to the calculations.
  
  So that every integration can be calculated using these rules.
  
  In the lecture, they only listed 4 of the safe transformations, 3 of the 12 heuristic transformations, and 3 of the 26 anchors.
  
  In order to implement the full symbolic integration, I need the rules. So; what are those? or where can I find them?
  
  **Tags:** reference-request, integration
  > **Comments:**
  > **username_0**: If this does not suit this site, let me know.  
  > **username_1**: Hello, welcome to AI stack exchange. Please provide us with more information about the problem. Update with the necessary links and provide more content for the experts to understand in detail.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2022-01-10. Score: 2 | Accepted answer
  
  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**](https://en.wikipedia.org/wiki/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**](https://en.wikipedia.org/wiki/And%E2%80%93or_tree).
  
  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](https://dl.acm.org/doi/10.1145/321186.321193) (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](https://dspace.mit.edu/bitstream/handle/1721.1/11997/31225400-MIT.pdf?sequence=2).
  
  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](https://www.mathsisfun.com/calculus/integration-rules.html), [this](https://math24.net/indefinite-integral-basic-rules-integration.html), and [this](https://en.wikipedia.org/wiki/Lists_of_integrals)) that describe the common integration rules.
  
  > **Comments:**
  > **username_0**: That is the answer to my question. I've looked at some of the websites, however, I was looking for the list. My flaw was not looking for a paper! Thanks.  
  > **username_3**: Some interesting symbolic methods can also be found in the papers citing the Slagle's paper: [Citing papers](https://dl.acm.org/action/ajaxShowCitedBy?doi=10.1145/321186.321193)  
  
  ---
  
  