The symbolic integration methods are undocumented.
The integration methods are very likely table driven: some pattern matching is done and various integration methods are attempted until eventually one works (or none of them work.)
Forming the grobnar basis is one of the potential steps.
For what it is worth, you can examine code:
regexprep(char(evalin(symengine, 'expose(int)')), '\\n', '\n')
ans =
'proc(f, x = null())
name int;
local integral, xx, a, b, prop, unevaluated_int, options, optseq, u, errtext;
option hold;
begin
if args(0) = 0 then
error(message(\"symbolic:int:NoArgument\"))
end_if;
x := context(x);
optseq := context(args(3..args(0)));
if testtype(x, \"_exprseq\") then
optseq := (x[2..-1], optseq);
x := x[1]
end_if;
options := intlib::getOptions(optseq);
if options[Hold] then
return(hold(int)(context(f), x, optseq))
end_if;
if type(x) = \"_equal\" && type(rhs(x)) = \"_range\" && type(lhs(x)) ~= \"_mult\" then
xx := op(x, 1);
if domtype(xx) ~= DOM_IDENT then
error(message(\"symbolic:int:VariableMustBeIdentifier2\"))
end_if;
a := op(op(x, 2), 1);
b := op(op(x, 2), 2);
if protected(xx) ~= None then
develinfo(1, \"\".xx.\" is protected, can't do assumptions early on\")
else
if options[hold(Assumptions)] ~= FALSE then
if is(a <= b) = UNKNOWN && intlib::printWarningsFlag && options[NoWarning] ~= TRUE then
warning(message(\"symbolic:int:Assuming\", a <= b))
end_if;
prop := Dom::Interval([a, b]);
if type(prop) = Dom::Interval || prop = R_ then
if property::hasprop(xx) then
if intlib::printWarningsFlag && options[NoWarning] ~= TRUE && getprop(xx) ~= prop then
warning(message(\"symbolic:int:ChangingProperties2\", xx, prop, getprop(xx)))
end_if;
unassume(xx)
end_if;
if is(b < a) = TRUE then
assumeAlso(xx in Dom::Interval([b, a]))
else
if is(a <= b) = TRUE then
assumeAlso(xx in Dom::Interval([a, b]))
else
assumeAlso(xx in R_)
end_if
end_if
end_if
end_if
end_if
else
if type(x) = \"_equal\" && type(lhs(x)) = \"_mult\" then
xx := op(x, 1);
a := op(op(x, 2), 1);
b := op(op(x, 2), 2);
[u, xx] := split(xx, unit::isUnit)[1..2];
return(context(hold(int)(f, xx = a/u..b/u, options))*u)
else
if type(x) = \"_mult\" then
[u, x] := split(x, unit::isUnit)[1..2];
return(context(hold(int)(f, x, options))*u)
end_if
end_if
end_if;
f := hold(hold)(context(f));
if testargs() && has(context([args()]), FAIL) then
error(message(\"symbolic:int:InvalidArgument\"))
end_if;
if has(context([args()]), undefined) then
return(undefined)
end_if;
assert(op(f, 0) = hold(hold));
f := op(f);
if f::dom::int ~= FAIL then
return(f::dom::int(f, x, context(args(3..args(0)))))
else
if domtype(f) = DOM_EXPR && domtype(eval(op(f, 0))) = DOM_FUNC_ENV && (eval(op(f, 0)))::int ~= FAIL then
return((eval(op(f, 0)))::int(f, x, context(args(3..args(0)))))
else
if testtype(f, Type::Arithmetical) = FALSE then
error(message(\"symbolic:int:InvalidIntegrand\"))
end_if
end_if
end_if;
if args(0) = 1 then
error(message(\"symbolic:int:MissingSecondArgument\"))
end_if;
if testtype(x, \"_exprseq\") then
options := intlib::getOptions(x[2..-1], options);
x := op(x, 1)
end_if;
if domtype(x) = DOM_IDENT then
integral := hold(int)(f, x, context(args(3..args(0))));
traperror((integral := intlib::int(f, x, options)));
integral := subs(integral, hold(intlib::frozenInt) = hold(int), Unsimplified);
unevaluated_int := FALSE;
if domtype(integral) ~= piecewise || domtype(piecewise::disregardPoints(integral)) ~= piecewise then
misc::maprec(piecewise::disregardPoints(integral), {\"int\"} = (i -> (if op(i, [2, 1]) = op(DOM_VAR(1,13), 1) then
DOM_VAR(1,7) := TRUE;
misc::breakmap()
end_if;
i)), Unsimplified)
end_if;
if unevaluated_int && ~options[hold(AlwaysExtractIntegrableParts)] = TRUE && 100 + intlib::Simplify::defaultValuation(hold(int)(f, x)) < intlib::Simplify::defaultValuation(integral) then
return(hold(int)(f, x, intlib::printOptions(options, x)))
end_if;
if ~unevaluated_int && hastype(integral, piecewise) then
integral := subs(integral, hold(int) = hold(intlib::frozenInt), Unsimplified);
integral := misc::maprec(integral, {piecewise} = (p -> piecewise::mapConditions(p, simplify::simplifyCondition))) assumingAlso f in C_;
integral := subs(integral, hold(intlib::frozenInt) = hold(int), Unsimplified)
end_if;
return(integral)
else
if type(x) = \"_equal\" && type(op(x, 2)) = \"_range\" then
intlib::xrngTest(x);
integral := hold(int)(f, x, context(args(3..args(0))));
if traperror((integral := intlib::defInt(f, x, options))) = 1028 then
errtext := getlasterror()[2];
if stringlib::contains(errtext, (message(\"symbolic:int:InvalidBorder\"))::getString()) then
error(message(\"symbolic:int:InvalidBorder\"))
end_if;
if stringlib::contains(errtext, (message(\"symbolic:int:DifferentRealParts\"))::getString()) then
error(message(\"symbolic:int:DifferentRealParts\"))
end_if
end_if;
if ~hastype(integral, \"int\") && ~hastype(integral, \"limit\") then
integral := intlib::Simplify(integral)
end_if;
integral := subs(integral, hold(intlib::frozenInt) = hold(int), Unsimplified);
unevaluated_int := FALSE;
if domtype(integral) ~= piecewise then
misc::maprec(integral, {\"int\"} = (i -> (if op(i, [2, 1]) = op(DOM_VAR(1,13), 1) then
DOM_VAR(1,7) := TRUE;
misc::breakmap()
end_if;
i)), Unsimplified)
end_if;
if unevaluated_int && ~options[hold(AlwaysExtractIntegrableParts)] = TRUE && 3*intlib::Simplify::defaultValuation([f, x]) + 200 < intlib::Simplify::defaultValuation(integral) then
return(hold(int)(f, x, intlib::printOptions(options, x)))
end_if;
return(integral)
else
error(message(\"symbolic:int:InvalidSecondArgument\"))
end_if
end_if
end_proc'
