Equation with table below

I want a table to show the process involved in integration by parts. Here is what I have (from Equation with vertical arrow):

\documentclass{article}
\usepackage{mathtools}

\newcommand\vertarrowbox[3][6ex]{%
\begin{array}[t]{@{}c@{}} #2 \\
\left\uparrow\vcenter{\hrule height #1}\right.\kern-\nulldelimiterspace\\
\makebox[0pt]{\normalsize#3}
\end{array}%
}
\begin{document}

\begin{align*}
\displaystyle \int x\exp(ax)\, dx\, &\vertarrowbox[8pt]{=}{%
\begin{tabular}{|l|l|}
\hline
$u=x$ & $du=dx$ \\\hline
$dv=\exp(ax)\,dx$ & $v=\displaystyle\frac{1}{a}\exp(ax)$\\\hline
\end{tabular}} \frac{x}{a}\exp(ax)-\frac{1}{a}\int\exp(ax)\, dx\\
&=\frac{1}{a}\exp(ax)\left(x-\frac{1}{a}\right)+c.
\end{align*}
\end{document}


I want to align the vertical line of the table with the arrow, and also if possible change the arrow style (like -Latex style in tikz).

I propose this solution, using mathtools, makebox and makecell, with some manual adjustment:

\documentclass{article}
\usepackage{mathtools, nccmath}
\usepackage{makebox, array, makecell}

\begin{document}

\bgroup
\setcellgapes{2pt}\makegapedcells
\begin{align*}
\int x\exp(ax)\, dx\, & = \frac{x}{a}\exp(ax)-\frac{1}{a}\int\exp(ax)\, dx\\
& \makebox*{${}={}$}{$\uparrow$}\\[-0.5ex]
\MoveEqLeft[-0.2] \mathclap{\begin{array}{|l|l|}
\hline
u=x & du=dx \\\hline
dv=\exp(ax)\,dx & v=\mfrac{1}{a}\exp(ax)\\\hline
\end{array}}\\
&=\frac{1}{a}\exp(ax)\biggl(x-\frac{1}{a}\biggr)+c.
\end{align*}
\egroup

\end{document}