How to temporarily escape an 'alignedat' environment nested inside an 'align*' environment?

Question

This is my output:

The problem, of course, is with the misplaced third line. What should happen is this:

• The four left-hand equal signs ought to be aligned;
• additionally minus signs and plus signs from the second and third lines should be aligned, respectively;
• this additional alignment should not affect first or fourth lines.

Here is what I tried (full working example simplified document, so you may copy):

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[LGR,T1]{fontenc}
\usepackage{ae,aecompl}
\usepackage[estonian]{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[a4paper]{geometry}
\usepackage{enumitem}
\usepackage{mathtools}
\usepackage[pdftex]{graphicx}

\newgeometry{margin=2.5cm}

\DeclareSymbolFont{upgreek}{LGR}{cmr}{m}{n}
\DeclareMathSymbol{\ppi}{\mathord}{upgreek}{`p}

\newcommand{\f}[2]{\frac{#1}{#2}}
\newcommand{\Int}[3]{\int\limits_{#1}^{#2}{#3}\, \mathrm{d}x}

\setlength\parindent{0pt}

\begin{document}

    \begin{align*}
                a_k &= \f{2}{\ppi}\Int{0}{\ppi}{f(x)} = \f{2}{\ppi}\Int{0}{\f{\ppi}{2}}{\left(
                \f{\ppi}{2}-x\right)} + \f{2}{\ppi}\Int{\f{\ppi}{2}}{\ppi}{\ppi} = \\
                &=\begin{alignedat}[t]{2}
                    \f{2}{\ppi} \f{\ppi}{2} \left(\f{\ppi}{2}\right)-&
                    \f{2}{\ppi} \f{1}{2} \left(\f{\ppi}{2}\right)^2 +& 
                    \f{2}{\ppi}\ppi\left(\ppi-\f{\ppi}{2}\right)=\\
                    &= \f{\ppi}{2} -& \f{\ppi}{4} +& 2\f{\ppi}{2} =
                \end{alignedat}\\
                &=\f{5}{4}\ppi
            \end{align*}

\end{document}

The part that is of interest to this question, of course, is the following subpart of the above:

\begin{document}

    \begin{align*}
                a_k &= \f{2}{\ppi}\Int{0}{\ppi}{f(x)} = \f{2}{\ppi}\Int{0}{\f{\ppi}{2}}{\left(
                \f{\ppi}{2}-x\right)} + \f{2}{\ppi}\Int{\f{\ppi}{2}}{\ppi}{\ppi} = \\
                &=\begin{alignedat}[t]{2}
                    \f{2}{\ppi} \f{\ppi}{2} \left(\f{\ppi}{2}\right)-&
                    \f{2}{\ppi} \f{1}{2} \left(\f{\ppi}{2}\right)^2 +& 
                    \f{2}{\ppi}\ppi\left(\ppi-\f{\ppi}{2}\right)=\\
                    &= \f{\ppi}{2} -& \f{\ppi}{4} +& 2\f{\ppi}{2} =
                \end{alignedat}\\
                &=\f{5}{4}\ppi
            \end{align*}

\end{document}

I believe I understand what the problem is. Have a look at the penultimate line inside the alignedat environment. There I have written &=; unfortunately, the line break \\ from the line previous only applies to the alignedat environment. It does not reach the align* environment.

So ideally this \\ should have a dual role, signalling alignedat and align* simultaneously. Simply put, can I do something like this (cf. \insertcommand{}):

\begin{document}

    \begin{align*}
                a_k &= \f{2}{\ppi}\Int{0}{\ppi}{f(x)} = \f{2}{\ppi}\Int{0}{\f{\ppi}{2}}{\left(
                \f{\ppi}{2}-x\right)} + \f{2}{\ppi}\Int{\f{\ppi}{2}}{\ppi}{\ppi} = \\
                &=\begin{alignedat}[t]{2}
                    \f{2}{\ppi} \f{\ppi}{2} \left(\f{\ppi}{2}\right)-&
                    \f{2}{\ppi} \f{1}{2} \left(\f{\ppi}{2}\right)^2 +& 
                    \f{2}{\ppi}\ppi\left(\ppi-\f{\ppi}{2}\right)=\\
                    \insertcommand{\\ &=} \f{\ppi}{2} -& \f{\ppi}{4} +& 2\f{\ppi}{2} =
                \end{alignedat}\\
                &=\f{5}{4}\ppi
            \end{align*}

\end{document}

so that the \\ &= is sent to align, and after that, alignedat continues?

I am very much hoping that such a minor tweak suffices; but other answers are welcome, too (if they do not require (much) manual usage of \!-s and so on).

Answer 1

If you want proper (horizontal) alignment across elements within the equation, you can use the measuring capabilities of eqparbox's \eqmakebox[<tag>][<align>]{<stuff>}. All elements with the same <tag> will be put in a box of maximum width (with an optional <align>ment; default is centred):

\documentclass{article}

\usepackage{amsmath,eqparbox}

\newcommand{\f}[2]{\frac{#1}{#2}}
\newcommand{\Int}[3]{\int\limits_{#1}^{#2}{#3}\, \mathrm{d}x}

\let\ppi\pi

\begin{document}

\begin{align*}
  a_k &= \f{2}{\ppi} \Int{0}{\ppi}{f(x)} 
    = \f{2}{\ppi} \Int{0}{\f{\ppi}{2}}{\Bigl( \f{\ppi}{2} - x \Bigr)} 
      + \f{2}{\ppi} \Int{\f{\ppi}{2}}{\ppi}{\ppi} \\
   &= \eqmakebox[first]{$\displaystyle \underbrace{\f{2}{\ppi} \f{\ppi}{2} \Bigl( \f{\ppi}{2} \Bigr)}$}
    - \eqmakebox[second]{$\displaystyle \underbrace{\f{2}{\ppi} \f{1}{2} \Bigl( \f{\ppi}{2} \Bigr)^2}$}
    + \eqmakebox[third]{$\displaystyle \underbrace{\f{2}{\ppi} \ppi \Bigl( \ppi - \f{\ppi}{2} \Bigr)}$} \\
   &= \eqmakebox[first]{$\displaystyle \f{\ppi}{2}$} 
    - \eqmakebox[second]{$\displaystyle \f{\ppi}{4}$} 
    + \eqmakebox[third]{$\displaystyle 2 \f{\ppi}{2}$} \\
   &= \f{5}{4} \ppi
\end{align*}

\end{document}