%%
%% The LaTeX Companion, 3ed
%%
%% Example 11-2-9 on page II-137 in "A single equation on several lines: with alignment".
%%
%% Copyright (C) 2022 Frank Mittelbach
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3c
%% of this license or (at your option) any later version.
%%
%% See https://www.latex-project.org/lppl.txt for details.
%%

\documentclass{tlc3examargin}
\pagestyle{empty}
\setcounter{page}{6}
\setlength\textwidth{330.59894pt}

  % Shorten the vertical size of the example ...
  \AtBeginDocument{%
    \setlength\abovedisplayskip{.2\abovedisplayskip}%
    \setlength\belowdisplayskip{.2\belowdisplayskip}%
  }

%StartShownPreambleCommands
\usepackage{amsmath}
\newcommand\relphantom[1]{\mathrel{\phantom{#1}}}
\newcommand\ve{\varepsilon}  \newcommand\tve{t_{\varepsilon}}
\newcommand\vf{\varphi}      \newcommand\yvf{y_{\varphi}}
\newcommand\bfE{\mathbf{E}}
%StopShownPreambleCommands

\begin{document}
\begin{equation}
  \begin{split}
    f_{h, \ve}(x, y)
      &= \ve \bfE_{x, y} \int_0^{\tve} L_{x, \yvf(\ve u)} \vf(x) \,du  \\
      &= h \int L_{x, z} \vf(x) \rho_x(dz)                             \\
      &\relphantom{=} {} + h \biggl[
         \frac{1}{\tve}
         \biggl( \bfE_{y} \int_0^{\tve}  L_{x, y^x(s)} \vf(x) \,ds
                 - \tve \int L_{x, z} \vf(x) \rho_x(dz)      \biggr) + \\
      &\relphantom{=} \phantom{{} + h \biggl[ }
         \frac{1}{\tve}
         \biggl( \bfE_{y} \int_0^{\tve}  L_{x, y^x(s)} \vf(x) \,ds
                 - \bfE_{x, y} \int_0^{\tve} L_{x, \yvf(\ve s)}
                                            \vf(x) \,ds    \biggr) \biggr]
  \end{split}
\end{equation}
\end{document}