Tortoise coordinate of a five-dimensional charged black hole ($\frac{d{r}_{\star }}{dr}=\frac{1}{f}$)

For distinct horizon radii $a>b>0$, integrating $d{r}_{\star }/dr=f^{-1}$ gives $r_{\star }=r+\frac{a^3}{2(a^2-b^2)}\log |(r-a)/(r+a)|-\frac{b^3}{2(a^2-b^2)}\log |(r-b)/(r+b)|+C$. The tortoise coordinate defines constant retarded or advanced coordinates on radial null geodesics.