The Easy Way
As egreg wrote, the easy answer is to \usepackage[upint]{stix}. XITS is a fork of the STIX font.
\documentclass{article}
\usepackage[upint]{stix}
\begin{document}
\[ \int \oint \oiint \iint \iiint \]
\end{document}

The Complicated Way
If you want to add individual symbols the legacy way, you need to look up in the package documantation which slot of which font they are in. The symbol you are looking for is on page 39, in slot "E8 of font stix-mathcal. Then you open up the file stix.sty and reverse-engineer the necessary setup for that symbol font.
Since there are two versions of the symbol, a display-style one in slot "E8 and a smaller one at "B2, you in fact would want to select between them with \mathchoice.
\documentclass{article}
\DeclareFontEncoding{LS2}{}{\noaccents@}
\DeclareFontSubstitution{LS2}{stix}{m}{n}
\DeclareSymbolFont{integrals}{LS2}{stixcal}{m}{n}
\DeclareMathSymbol{\ointupbig}{\mathop}{integrals}{"E8}
\DeclareMathSymbol{\ointupsmall}{\mathop}{integrals}{"B2}
\DeclareRobustCommand{\ointup}{\mathchoice{\ointupbig}{\ointupsmall}{\ointupsmall}{\ointupsmall}}
\begin{document}
\( \displaystyle
\oint_{\oint_{\oint}} \quad
\ointup_{\ointup_{\ointup}} \)
\( \oint \quad
\ointup \)
\end{document}

I do not recommend this, as it depends on fiddly little implementation details of the stix package, runs into the limit on legacy math alphabets, and has other quirks.
But, I think that was what you were asking how to do?
The Modern Way
You can also do this with unicode-math, and I recommend that unless you are forced to submit to a site that does not yet support LuaLaTeX or XeLaTeX.
If you want to use XITS Math as your math font, you can load it with
\setmathfont[StylisticSet=8]{XITS Math}
If you want to use the XITS upright integrals together with another font, you can add the command
\setmathfont[range={"222B-"2233,"2A0B-"2A1C},StylisticSet=8]{XITS Math}
after you load your main \setmathfont. This also works with STIX Two Math. If you’re mixing different math fonts with different heights, you might want to add the Scale=MatchUppercase font option.
\usepackage[upint]{stix}not produce the vertical closed integrals? If so, that appears to be a bug, and should be reported. – barbara beeton Aug 15 '19 at 16:44stix. Then you might try the techniques shown in one of these questions: Importing a single symbol from a different font or How to get only one symbol from a symbol package?. – barbara beeton Aug 15 '19 at 17:23\setmathfont[StylisticSet=8]{XITS Math}? – Mico Aug 15 '19 at 19:48\ointup, or\let\oint\ointup, right? Maybe you could what you want by giving a small piece of your code. – timothymctim Aug 15 '19 at 21:46