$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\downarrow}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,{\rm e}^{#1}\,}
\newcommand{\fermi}{{\rm f}}
\newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}
\newcommand{\half}{{1 \over 2}}
\newcommand{\ic}{{\rm i}}
\newcommand{\iff}{\Longleftrightarrow}
\newcommand{\imp}{\Longrightarrow}
\newcommand{\isdiv}{\,\left.\right\vert\,}
\newcommand{\ket}[1]{\left\vert #1\right\rangle}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\pars}[1]{\left(\, #1 \,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\pp}{{\cal P}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,}
\newcommand{\sech}{\,{\rm sech}}
\newcommand{\sgn}{\,{\rm sgn}}
\newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
\newcommand{\ul}[1]{\underline{#1}}
\newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}
\newcommand{\wt}[1]{\widetilde{#1}}$
$\ds{\large\tt\mbox{There is a list of}}$ $\quad\large\mbox{Vector Identities in Wikipedia}$ !!!.
\begin{align}
\nabla\times\vec{\fermi}\pars{\vec{r},t}&=
\nabla\times\bracks{%
{\expo{\ic\pars{cr- d t}} \over r}\,\vec{b}\times{\vec{r} \over r}}
\\[3mm]&=\nabla\bracks{{\expo{\ic\pars{cr- d t}} \over r^{2}}}\times
\pars{\vec{b}\times\vec{r}}
+{\expo{\ic\pars{cr- d t}} \over r^{2}}\nabla\times\pars{\vec{b}\times\vec{r}}
\\[3mm]&=\braces{{\vec{r} \over r}\,
\partiald{}{r}\bracks{{\expo{\ic\pars{cr- d t}} \over r^{2}}}}\times
\pars{\vec{b}\times\vec{r}}
+{\expo{\ic\pars{cr- d t}} \over r^{2}}\nabla\times\pars{\vec{b}\times\vec{r}}
\\[3mm]&={\phi_{\rm r} \over r}\,\,
\color{#44f}{\vec{r}\times\pars{\vec{b}\times\vec{r}}}
+ \phi\ \color{#c00000}{\nabla\times\pars{\vec{b}\times\vec{r}}}
\quad\mbox{where}\quad\phi\equiv{\expo{\ic\pars{cr- d t}} \over r^{2}}
\qquad\quad\pars{1}
\end{align}
$$
\color{#44f}{\vec{r}\times\pars{\vec{b}\times\vec{r}}}
=\vec{b}\pars{\vec{r}\cdot\vec{r}} - \vec{r}\pars{\vec{b}\cdot\vec{r}}
=r^{2}\,\vec{b} - \pars{\vec{b}\cdot\vec{r}}\vec{r}\tag{2}
$$
\begin{align}
&\color{#c00000}{\nabla\times\pars{\vec{b}\times\vec{r}}}
=\vec{b}\ \overbrace{\nabla\cdot\vec{r}}^{\ds{=\ 3}}\ -\
\vec{r}\ \overbrace{\nabla\cdot\vec{b}}^{\ds{=\ 0}}\ +\ \overbrace{\pars{\vec{r}\cdot\nabla}\vec{b}}^{\ds{=\ 0}} - \pars{\vec{b}\cdot\nabla}\vec{r}
\\[3mm]&\mbox{and}\quad\pars{\vec{b}\cdot\nabla}\vec{r}
=\sum_{i}b_{i}\,\partiald{}{x_{i}}\sum_{j}\hat{e}_{j}x_{j}
=\sum_{ij}b_{i}\hat{e}_{j}\delta_{ij}=\sum_{i}b_{i}\hat{e}_{i}=\vec{b}.
\\[3mm]&\mbox{Then,}\quad\color{#c00000}{\nabla\times\pars{\vec{b}\times\vec{r}}}
=2\vec{b}\tag{3}
\end{align}
With $\pars{2}$ and $\pars{3}$, $\pars{1}$ is reduced to:
\begin{align}
\nabla\times\vec{\fermi}\pars{\vec{r},t}&=
{\phi_{\rm r} \over r}\bracks{r^{2}\,\vec{b} - \pars{\vec{b}\cdot\vec{r}}\vec{r}}
+\phi\pars{2\vec{b}}
\end{align}
\begin{align}&\color{#44f}{\large%
\nabla\times\vec{\fermi}\pars{\vec{r},t}=
\pars{2\phi + r\phi_{\rm r}}\vec{b}
-{\vec{b}\cdot\vec{r} \over r}\,\phi_{\rm r}\,\vec{r}}
\\[3mm]&\mbox{with}\quad\phi\equiv{\expo{\ic\pars{cr- d t}} \over r^{2}}
\quad\mbox{and}\quad\phi_{\rm r} = \partiald{\phi}{r}.
\end{align}