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

\newcommand\testsb[1]{%
\tikz[line width=0.1pt,baseline]{
\node[anchor=base,inner sep=0pt] (X) {$#1$} ;
\draw[gray] (X.north east) rectangle (X.south west)
(X.north east)--(X.south west) (X.north west)--(X.south east)
(X.north)--(X.south) (X.west)--(X.east);
}}

\newsavebox\intbox
\newsavebox\intboxx
\newcommand\intjq[5]{%
\sbox\intbox{$#1#2#5$}\sbox\intboxx{$#1#3#5$}%
\tikz[baseline={(0,0)},overlay,line width=0.1ex]{
\draw (\wd\intbox-0.5\wd\intboxx,0.5\ht\intbox-0.5\dp\intbox-#4\wd\intboxx)
arc [start angle=-90, end angle=90, radius=#4\wd\intboxx]
--++(\wd\intboxx-\wd\intbox,0)
arc [start angle=90, end angle=270, radius=#4\wd\intboxx]
--cycle;
}\!#2}

\newcommand\kkoint{\intjq{\displaystyle}{\int}{\int}{0.5}{\mkern0.3mu}}
\newcommand\kkoiint{\intjq{\displaystyle}{\iint}{\int}{0.5}{\mkern0.3mu}}
\newcommand\kkoiiint{\intjq{\displaystyle}{\iiint}{\int}{0.5}{\mkern0.3mu}}

\newcommand\kkoidotsint{\intjq{\displaystyle}{\idotsint}{\int}{0.5}{\mkern0.3mu}}

\newcommand\kktoint{\intjq{}{\int}{\int}{0.3}{\mkern0.1mu}}
\newcommand\kktoiint{\intjq{}{\iint}{\int}{0.3}{\mkern0.1mu}}
\newcommand\kktoiiint{\intjq{}{\iiint}{\int}{0.3}{\mkern0.1mu}}

\begin{document}

\[
\int\quad
\iint\quad
\iiint\quad
\idotsint\quad
\oint\quad
\oiint\quad
\oiiint
\]

\[
\testsb{\displaystyle\int}\quad
\testsb{\displaystyle\int\mkern0.3mu}
\]

\[
\kkoint\quad
\kkoiint\quad
\kkoiiint\quad
\kkoidotsint
\]

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

$\kktoint$
$\kktoiint$
$\kktoiiint$

\end{document}