Background color for different terms in equation

Question

How to put a different background color for different terms in one single equation. Below is the example I tried. I need boxed term in the equation to have a background of green, the second term with magenta, and the last term to be a yellow background. how to achieve this. The expected result looks like the last equation.

\documentclass{article}
\pagestyle{empty}
\usepackage{amsmath}

\newcommand{\diff}{\mathop{}\!\mathrm{d}}

\documentclass{article}
\pagestyle{empty}
\usepackage{amsmath}

\usepackage{empheq}
\usepackage{xcolor}
\definecolor{lightgreen}{HTML}{90EE90}
\newcommand{\boxedeq}[2]{\begin{empheq}[box={\fboxsep=6pt\fbox}]{align}\label{#1}#2\end{empheq}}
\newcommand{\coloredeq}[2]{\begin{empheq}[box=\colorbox{lightgreen}]{align}\label{#1}#2\end{empheq}}

\newcommand{\diff}{\mathop{}\!\mathrm{d}}

\begin{document}
\begin{subequations}
 \begin{align}
 \mathcal{\bar{L}}&=\boxed{\textcolor{red}{\frac{\beta^4}{2}\left(\int_{0}^{1}\,\hspace{-0.05in} \left( \psi(\eta)\right)^2 \diff\eta + \sum_{j=1}^{n} \gamma_{j}^3\, \int_{0}^{\gamma_{j}}  \hspace{-0.05in} \left( \chi_{j}(\xi)\right)^2 \diff\xi \right)}}-\nonumber \\ &\quad  \textcolor{blue}{\frac{1}{2}\,\left(\int_{0}^{1} \hspace{-0.05in} \left(\frac{\partial^2\psi(\eta)}{\partial \eta^2}\right)^2 \diff\eta  +\sum_{j=1}^{n}\alpha_{j}\, \int_{0}^{\gamma_{j}}\left(\frac{\partial \chi_{j}(\xi)}{\partial \xi}\right)^2 \diff\xi \right)}+\nonumber \\ &\quad \sum_{j=1}^{n}\lambda_{j}^{*}\left(\psi(\zeta_{j})-\chi_{j}(\gamma)\right) \nonumber  
 \end{align}
 \end{subequations}
\coloredeq{eq:second}{c_i = \langle\psi|\phi\rangle} 
\end{document}

Answer 1

The operative part is given below -- please see if it meets the rquirement

          \tikz[baseline]{\node[fill=green!25, rectangle, anchor=base]
                {.............}%<-------------------------------item to be colored 
                }

    \documentclass{article}
\pagestyle{empty}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{shapes}

\newcommand{\diff}{\mathop{}\!\mathrm{d}}

\begin{document}
\begin{subequations}
 \begin{align}
 \mathcal{\bar{L}}&=\boxed{\textcolor{red}{\frac{\beta^4}{2}\left(\int_{0}^{1}\,\hspace{-0.05in} \left( \psi(\eta)\right)^2 \diff\eta + \sum_{j=1}^{n} \gamma_{j}^3\, \int_{0}^{\gamma_{j}}  \hspace{-0.05in} \left( \chi_{j}(\xi)\right)^2 \diff\xi \right)}}-\nonumber \\ &\quad  \textcolor{blue}{\frac{1}{2}\,\left(\int_{0}^{1} \hspace{-0.05in} \left(\frac{\partial^2\psi(\eta)}{\partial \eta^2}\right)^2 \diff\eta  +\sum_{j=1}^{n}\alpha_{j}\, \int_{0}^{\gamma_{j}}\left(\frac{\partial \chi_{j}(\xi)}{\partial \xi}\right)^2 \diff\xi \right)}+\nonumber \\ &\quad \sum_{j=1}^{n}\lambda_{j}^{*}\left(\psi(\zeta_{j})-\chi_{j}(\gamma)\right) \nonumber  
 \end{align}
 \end{subequations}
{eq:second} \tikz[baseline]{
                \node[fill=green!25, rectangle, anchor=base]
                {$c_i = \langle\psi|\phi\rangle$} 
                            }
\end{document}

Answer2

The other option is to use \colorbox, provided by the package xcolor. Here's the result (along with some streamlining of your code and replacing \left and \right directives with explicit sizing instructions):

\documentclass{article}
\usepackage{xcolor,amsmath,tikz}
\usetikzlibrary{shapes}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}

\begin{document}
%\begin{subequations}
\begin{align*}
\colorbox{green}{$\mathcal{\bar{L}}$}
   &=\boxed{%
   \textcolor{red}{ \frac{\beta^4}{2} \biggl(\,
   \colorbox{green}{$\displaystyle \int_{0}^{1} 
   ( \psi(\eta))^2 \diff\eta$} 
   + \sum_{j=1}^{n} \gamma_{j}^3 \int_{0}^{\gamma_{j}}  
   ( \chi_{j}(\xi) )^2 \diff\xi 
   \biggr) } % end \textcolor
   }  % end \boxed
   \\ 
   &\quad - 
   \textcolor{blue}{ \frac{1}{2} \biggl( 
   \int_{0}^{1} \biggl(
   \frac{\partial^2\psi(\eta)}{\partial \eta^2}\biggr)^{\!2} \diff\eta  
   +\sum_{j=1}^{n}\alpha_{j} \! \int_{0}^{\gamma_{j}}
   \biggl(\frac{\partial \chi_{j}(\xi)}{\partial \xi}\biggr)^{\!2} 
   \diff\xi \biggr)}\\ 
   &\quad +
   \sum_{j=1}^{n}\lambda_{j}^{*} \bigl(\psi(\zeta_{j})-\chi_{j}(\gamma)\bigr)  
\end{align*}
%\end{subequations}
{eq:second} \tikz[baseline]{
                \node[fill=green!25, rectangle, anchor=base]
                {$c_i = \langle\psi|\phi\rangle$} 
                            }
\end{document}