integration by parts. The second-order Lagrangian ${\cal L}^{(2)}$ is of the form $${\cal L}^{(2)}=a_1 h_0^2+a_2 h_1^2+a_3 \Bigl[ {\dot h_1}^2+h_0'^2-2 {\dot h_1} h_0'+2\frac{C'}{C} {\dot h_1} h_0 \Bigr]\,, \label{lag-odd1}$$ where a dot represents differentiation with respect to $t$ and $'$ with respect to $r$. The coefficients $a_1, a_2$ and $a_3$ are given by $$\begin{aligned}
a_1&=\frac{\ell (\ell+1)}{4 C}
\Bigl[\frac{{\rm d}}{{\rm d} r}\left(
C'\sqrt{\frac{B}{A}}{\cal H}\right)
+\frac{(\ell-1) (\ell+2)}{\sqrt{AB}}{\cal F}\nonumber\\
&+\frac{2C}{\sqrt{AB}}  {\cal E}_A
\Bigr]\,,
\\
a_2&=- \frac{\ell (\ell+1)}{2} \sqrt{AB} \left[ \frac{(\ell-1) (\ell+2)}{2 C}{\cal G}+{\cal E}_B \right]\,,
\\
a_3&=
\frac{\ell (\ell+1)}{4} \sqrt{\frac{B}{A}} {\cal H}\,.\end{aligned}$$ On-shell, ${\cal E}_A={\cal E}_B=0$ and hence the above expressions simplify a bit $$\begin{aligned}
{\cal F}&=2\left(
G_4+\frac{1}{2}B\phi'X'G_{5X}-XG_{5\phi}
\right)\,, \label{def-f}
\\
{\cal G}&=2\left[ G_4-2XG_{4X}
+X\left(\frac{A'}{2A}B\phi'G_{5X}+G_{5\phi}\right)
\right]\,, \label{def-g}
\\
{\cal H}&=2\left[G_4-2X G_{4X}
+X\left(\frac{  C' }{2 C} B \phi' G_{5X}+ G_{5\phi}\right)\right]\,. \label{def-h}\end{aligned}$$ and therefore we recover the same results as [@Kobayashi:2012kh].

The Lagrangian (\[lag-odd1\]) can be simplified by noticing that the ${\dot h_0}$ is absent and therefore it is an auxiliary field. But it is complicated to integrate it out using its equation of motion because of the presence of the term $h_0'^2$ in the Lagrangian. Therefore, we introduce an additional field [@DeFelice:2011ka] by rewriting the Lagrangian in the following form $${\cal L}^{(2)}=\Bigl[ a_1-\frac{(C' a_3)'}{C} \Bigr] h_0^2
