\documentclass{article}
\usepackage[euler]{varint}
\usepackage{tikz}

\newsavebox\intbox
\newsavebox\intboxx
\newcommand\intjq[4]{\mathop{}\!%
\sbox\intbox{$#1#2#4$}\sbox\intboxx{$#1\int#4$}%
\tikz[baseline={(0,0)},overlay,line width=0.08\wd\intboxx]{
\draw (\wd\intbox-0.5\wd\intboxx,0.5\ht\intbox-0.5\dp\intbox-#3\wd\intboxx)
arc [start angle=-90, end angle=90, radius=#3\wd\intboxx]
--++(\wd\intboxx-\wd\intbox,0)
arc [start angle=90, end angle=270, radius=#3\wd\intboxx]
--cycle;
}\!#2}

\newcommand\kkoint{\intjq{\displaystyle}{\int}{0.5}{\mkern0.3mu}}
\newcommand\kkoiint{\intjq{\displaystyle}{\iint}{0.5}{\mkern0.3mu}}
\newcommand\kkoiiint{\intjq{\displaystyle}{\iiint}{0.5}{\mkern0.3mu}}
\newcommand\kktoint{\intjq{}{\int}{0.3}{\mkern0.1mu}}
\newcommand\kktoiint{\intjq{}{\iint}{0.3}{\mkern0.1mu}}
\newcommand\kktoiiint{\intjq{}{\iiint}{0.3}{\mkern0.1mu}}
\newcommand\kksoint{\intjq{\scriptstyle}{\int}{0.3}{}}
\newcommand\kksoiint{\intjq{\scriptstyle}{\iint}{0.3}{}}
\newcommand\kksoiiint{\intjq{\scriptstyle}{\iiint}{0.3}{}}
\newcommand\kkssoint{\intjq{\scriptscriptstyle}{\int}{0.3}{}}
\newcommand\kkssoiint{\intjq{\scriptscriptstyle}{\iint}{0.3}{}}
\newcommand\kkssoiiint{\intjq{\scriptscriptstyle}{\iiint}{0.3}{}}

\newcommand\Oint{\mathchoice{\kkoint}{\kktoint}{\kksoint}{\kkssoint}}
\newcommand\Oiint{\mathchoice{\kkoiint}{\kktoiint}{\kksoiint}{\kkssoiint}}
\newcommand\Oiiint{\mathchoice{\kkoiiint}{\kktoiiint}{\kksoiiint}{\kkssoiiint}}

\newcommand\kkoidotsint{\intjq{\displaystyle}{\idotsint}{0.5}{\mkern0.3mu}}
\newcommand\kktoidotsint{\intjq{}{\idotsint}{0.3}{\mkern0.1mu}}

\begin{document}

$\displaystyle\int\quad\iint\quad\iiint\quad\idotsint\quad\oint$
(Euler fonts original)

$\displaystyle\Oint\quad\Oiint\quad\Oiiint$
(TikZ draw $\intjq{\displaystyle}{\phantom\iint}{0.4}{}\,$s)
\quad$\displaystyle\kkoidotsint$ (just a test)

\vspace{10pt}

$\int\quad\iint\quad\iiint\quad\idotsint\quad\oint$

$\Oint\quad\Oiint\quad\Oiiint\quad\kktoidotsint$

\vspace{10pt}

$\begin{aligned}
k&=\int_u^ck & k&=\iint_u^ck & k&=\iiint_u^ck\\
k&=\kkoint_u^ck & k&=\kkoiint_u^ck & k&=\kkoiiint_u^ck\\
&k\int\limits_u^ck & &k\iint\limits_u^ck & &k\iiint\limits_u^ck\\
&k\kkoint\limits_u^ck & &k\kkoiint\limits_u^ck & &k\kkoiiint\limits_u^ck
\end{aligned}$

\vspace{10pt}

\newcommand\testsup[1]{#1\frac{#1^{#1^{#1}}}{#1_{#1_{#1}}}}
$\displaystyle\testsup{\Oint}\quad\testsup{\Oiint}\quad\testsup{\Oiiint}$

\end{document}
