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Documentation on replacement rules used in Integrate?

Posted 1 month ago
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I'm looking into polylogarithmic integrals and I've stumbled upon some integrals of the following forms:

Integrate[PolyLog[#, x]/(x + a), x] & /@ {0, 1, 2} // TableForm

Sadly Mathematica comes up empty for #=3 and as it turns out, the #=2 is merely looked up from a table of replacement rules.

To see why, I believe the RUBI would be the standard for tracing the integration process: https://rulebasedintegration.org - but this too simply points to a "Find-and-replace" approach: at some point in the integration the expression Integrate[log(x+a) log(x+b)/x,x] is simply replaced with its solution, with a proper derivation completely omitted.

My question is simple but the answer is likely not: where can I find a reference for the result of the following command,

Integrate[PolyLog[2, x]/(x + a), x]

which is:

Log[1 - x] Log[x] Log[a + x] + 
 1/2 (Log[x] - Log[-(x/a)]) Log[(a + x)/
   a] (-2 Log[1 - x] + Log[(a + x)/a]) + (-Log[x] + Log[-(x/a)]) Log[(
   a + x)/a] Log[(a + x)/(a - a x)] + 
 1/2 (Log[(1 + a)/(1 - x)] + Log[x] - 
    Log[((1 + a) x)/(a (-1 + x))]) Log[(a + x)/(
   a - a x)]^2 + (Log[a + x] - Log[(a + x)/(a - a x)]) PolyLog[2, 
   1 - x] + 
 Log[a + x] PolyLog[2, 
   x] + (Log[1 - x] + Log[(a + x)/(a - a x)]) PolyLog[2, (a + x)/a] + 
 Log[(a + x)/(
   a - a x)] (PolyLog[2, (a + x)/(-1 + x)] - 
    PolyLog[2, (a + x)/(a - a x)]) - PolyLog[3, 1 - x] - 
 PolyLog[3, (a + x)/a] - PolyLog[3, (a + x)/(-1 + x)] + 
 PolyLog[3, (a + x)/(a - a x)]

... or perhaps a list with references that Wolfram uses to prove the claim that Mathematica gives (without the obvious solution of just "taking the derivative" of the expression shown).

Alternatively, can anyone elucidate how to read the TraceInternal output from:

Trace[Integrate[PolyLog[2, x]/(x + a), x], "TraceInternal" -> True]
Posted 1 month ago

To add to the conversation: I have found at least something but no mention on the techniques employed to find the result. See page 36 on the pdf at the RUBI GitHub page .

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