The caption wasn't really centred (hard to tell as the text was short)
it is a justified paragraph (set in a centred box) the problem is that
bidi doesn't "patch" the caption code to add \beginR it redefines the
standard one, so over-writing the caption package changes.
The following simply preserves the longtable/caption package version
of \caption when bidi is loaded (so it doesn't get patched to right
to left usage, so you may need to add \beginR \endR explicitly in a
right to left caption, patching automatically might be tricky as you
have to cope with caption, ltcaption, longtable and bidi being loaded
or not, in different package loading orders.
\documentclass{report}
\usepackage{longtable}
%Tell the caption pkg to do ragged right at load time:
\usepackage[singlelinecheck=off
,justification=raggedright
]{caption}
\usepackage{ltcaption}
\LTcapwidth=\linewidth
\makeatletter
\let\temp\LT@makecaption
\usepackage{bidi}
\let\LT@makecaption\temp
%Tell the caption pkg again:
\captionsetup{singlelinecheck=off
,justification=raggedright
}
\begin{document}
\begin{table}
\caption{A normal table
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
}
\end{table}
\begin{longtable}{llll}
\caption{A long table
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
xxxx xxxx xxxx xxxx xxxx xxxxxxxx
}%Should be ragged right, but is centered instead
\tabularnewline
{AAAA} & {BBBB} & {CCCC} & {DDDD}
\tabularnewline
\end{longtable}
\end{document}
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