This worksheet provides the primitives of the functions $r\mapsto (r^2+a^2)/\Delta$ and $r\mapsto 1/\Delta$, which appear in the relations between Boyer-Lindquist coordinates and Kerr ones.
We set $m=1$.
%display latex
var('r a', domain='real')
assume(a<1)
assume(a>=0)
f = (r^2 + a^2)/(r^2 - 2*r + a^2)
f
s = integrate(f, r)
s
diff(s, r).simplify_full()
rp = 1 + sqrt(1-a^2)
rm = 1 - sqrt(1-a^2)
F = r + 1/sqrt(1-a^2)*(rp*ln(abs((r-rp)/2)) - rm*ln(abs((r-rm)/2)))
F
dFdr = diff(F,r).simplify_full()
dFdr
bool(dFdr == f)
g = 1/(r^2 - 2*r + a^2)
g
integrate(g,r)
G = 1/(2*sqrt(1-a^2))*ln(abs((r-rp)/(r-rm)))
G
dGdr = diff(G,r).simplify_full()
dGdr
bool(dGdr == g)