Sorry, if this question sounds a bit amateurish, but I haven't been able to find a good summary about the differences between these four environments for multi-lining an equation. Where do they differ and which one should I choose under different circumstances?

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    For an excellent summary of what the split, multline, and align environments do, I recommend reading section 3, "Displayed equations", of the user guide of the amsmath package. It may be loaded by typing texdoc amsmath at a command prompt. Similarly, to bring up the user guide of the breqn package, type texdoc breqn at a command prompt. – Mico Apr 18 '15 at 10:40
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    Also texdoc mathmode is a good source. – egreg Apr 18 '15 at 13:52

As noted in comments the amsmath and breqn documentation have several good examples, also the mathtools package has extended versions of several of the amsmath alignments. But the usual style here is to answer inline rather than refer to manuals, so this is a document giving the basic usage of the environments you mention.

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Align, from amsmath package:
numbered equations aligned at points marked
with \verb|&| usually just before a relation.
a_1& =b_1+c_1\\
a_2& =b_2+c_2-d_2+e_2

split, also from amsmath,
similar alignment to align, but the whole construct fits within
equation (or other display math) and is numbered as a unit.
a& =b+c-d\\
 & \quad +e-f\\
 & =g+h\\
 & =i

multline, from amsmath
for lonq expressions taking more than one line,
with no specifed alignment points.

breqn is not part of the amsmath collection and is a highly experimental
package that tries to automate the line breaking.
Here it automatically spots the relations and adds
the line breaks and alignment points.
It also automatically handles the trailing full stop after the display.
T(n) \hiderel{\leq} T(2^{\lceil\lg n\rceil})
  \leq c(3^{\lceil\lg n\rceil}
    -2^{\lceil\lg n\rceil})
  <3c\cdot3^{\lg n}

  • How you solve the same problem with parenthesis like \left[ and \right]? – Alejandro Sazo Sep 15 '16 at 18:50
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    @AlejandroSazo ? sorry I can't guess what problem you mean. Perhaps you should ask a question on the site. – David Carlisle Sep 15 '16 at 19:58
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    @Xaser no, perhaps I should have used \qquad the apparent alignment there was accidental and unintended – David Carlisle Sep 17 '18 at 17:19
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    @Xaser I don't know of any formal convention but the styles are normally to just wrap so not indented at all, but if you choose to indent it's better if it's obviously indented and not looking like it's aligned with something. \quad and \qquad are available for that use but I wouldn't say theer are really any conventions. – David Carlisle Sep 17 '18 at 17:25
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    @Xaser, here some notes about spacing in math mode: overleaf.com/learn/latex/Spacing_in_math_mode. I wanted to align a letter n (not a +), and what worked for me was something like &\quad\;. \quad for the space of the = that I am not putting in that line, and \; for the standard space after the = operator. – Carlos Mendoza Jun 12 '19 at 15:24

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