\frac{2BC'}{C}\left( 
\frac{2C''}{C'}-\dfrac{C'}{2C}+\frac{X'}{X}+\frac{B'}{B} \right) X G_{4X}
\nonumber\\&&
+\frac{4BC'}{C}XX'G_{4XX}-B\phi'\left(\frac{2C'}{C}+\frac{X'}{X}\right)
G_{4\phi}
+4X G_{4\phi\phi}
+2B\phi' \left(\frac{2C'}{C}-\frac{X'}{X} \right) X  G_{4\phi X}
\nonumber\\&&
+
\frac{B\phi' }{C}
\left[(1-\dfrac{3BC'^2}{4C})\frac{X'}{X}+\dfrac{BC'}{C}\left(\dfrac{C'^2}{2C}-C''\right)  -\dfrac{B'C'^2}{2C} \right] X G_{5X}
-\frac{B^2 C'^2 \phi'}{2C^2} XX'G_{5XX}
\nonumber\\&&
-
\frac{2}{C}\left[1+B\left(C'' -\frac{C'^2}{4C}\right)+BC'\frac{X'}{X}+\dfrac{B'C'}{2}
\right] X  G_{5\phi}
-\frac{2BC'\phi'}{C} XG_{5\phi\phi}
+\frac{2BC'}{C}\left(\frac{C'}{2C}-\frac{X'}{X} \right)X^2 
G_{5\phi X},
\\
{\cal E}_B&:=&K-2XK_X
+\left(\frac{2C'}{C}+\frac{A'}{A}\right)B \phi' XG_{3X}+2X G_{3\phi}
\nonumber\\&&
+\frac{2}{C}\left(1-\dfrac{BC'^2}{4C}-\dfrac{A'BC'}{2A}\right)G_4
-\frac{4}{C}\left(1-\dfrac{BC'^2}{2C}-\frac{A'BC'}{A}\right)XG_{4X}
+\frac{4BC'}{C}\left(\frac{C'}{2C}+\frac{A'}{A}\right)X^2G_{4XX}
\nonumber\\&&
-\left(\frac{2C'}{C}+ \frac{A'}{A}\right)B\phi'G_{4\phi}
-2\left(\frac{2C'}{C}+ \frac{A'}{A}\right)B\phi'XG_{4\phi X}
+\frac{B\phi'}{C}\left(1-\dfrac{5BC'^2}{4C}\right)\frac{A'}{A}XG_{5X}
\nonumber\\&&
-\frac{A'B^2C'^2 \phi'}{2AC^2}X^2G_{5XX}
+\frac{2}{C}\left(1-\dfrac{3BC'^2}{4C}-\dfrac{3A'BC'}{2A}\right)XG_{5\phi}
-\frac{2BC'}{C}\left(\frac{C'}{2C}+ \frac{A'}{A}\right)X^2G_{5\phi X},
\\
{\cal E}_C&:=&K-2XG_{3\phi}+B\phi' X' G_{3X} -\left[ \sqrt{\frac{B}{A}} \left( \sqrt{\frac{B}{A}} A' \right)'+\sqrt{\frac{B}{C}} \left( \sqrt{\frac{B}{C}} C' \right)'+\frac{A'BC'}{2AC} \right]G_4
\nonumber\\&&
+
\dfrac{2}{\phi'} \left(\frac{A'}{A} + \frac{C'}{C} + \frac{X'}{X} \right)
X G_{4\phi} 
+
B X \left( - \frac{A'^2}{A^2}- \frac{C'^2}{C^2}+\frac{A'B'}{AB}+\dfrac{A'C'}{AC}+\dfrac{B'C'}{BC} +\dfrac{2A''}{A}+\dfrac{2C''}{C} \right) 
\left( G_{4X}-\frac{1}{2}G_{5\phi} \right)
\nonumber\\&&
+
BX' \left( \frac{C'}{C}+\frac{A'}{A} \right) \left( G_{4X}-G_{5\phi} \right)+4X G_{4\phi\phi}
+ 2B\phi' \left( \frac{C'}{C} + \frac{A'}{A} - \frac{X'}{X} \right) X  G_{4\phi X}
\nonumber\\&&
+ 2B \left( \frac{C'}{C}+\frac{A'}{A} \right) XX' G_{4XX}
-
\frac{B^2\phi'}{2C} \left[\frac{A''C'}{A}-\frac{A'^2C'}{2A^2} + \frac{A'}{A} \left(C''-\dfrac{C'^2}{2C}+\frac{B'C'}{B}+\frac{3C'X'}{2X} \right) \right] X G_{5X}
\nonumber\\&&
-B\phi' X \left( \frac{C'}{C}+\frac{A'}{A} \right) G_{5\phi \phi}
+
B \left[\dfrac{A'C'}{AC}-\frac{X'}{X} \left( \frac{C'}{C}+\frac{A'}{A} \right) \right] X^2
G_{5\phi X}
\nonumber\\&&
-\frac{A'B^2C' \phi'}{2AC} XX' G_{5XX},\end{aligned}$$

[99]{} E. Berti [*et al.*]{}, Class. Quant. Grav.  [**32**]{} (2015) 243001 doi:10.1088/0264-9381/32/24/243001 \[arXiv:1501.07274 \[gr-qc\]\].

T. Regge and J. A. Wheeler, Phys. Rev.  [**108**]{} (1957) 1063. doi:10.1103/PhysRev.108.1063 F. J. Zerilli, Phys. Rev. 