apparently current verion (14) Mathematica has forgotten the needed integral

d phi/dr = \dot{phi}/\dot{r} = L/(r^2 \dot{r})

GR 3.147.7 p 242
Integrate[1/Sqrt[(a-x)(x-b)(x-c)(x-d)],{x,u,a]=2/Sqrt[(a-c)(b-d)] F[mu,r]
for us with d=0, mu=Pi/2, r=Sqrt[(a-b)c/((a-c)b)]
Integrate[1/Sqrt[(a-x)(x-b)(x-c)(x)],{x,b,a]=2/Sqrt[(a-c)(b)] E[r]

sigma[r_]=1-2/r
e2=Simplify[(rb^2-ra^2)/(rb^2/sigma[rb]-ra^2/sigma[ra])]
L2=Simplify[(sigma[rb]-sigma[ra])/(sigma[ra]/ra^2-sigma[rb]/rb^2)]
Simplify[e2-(L2/r^2+1)(1-2/r)]*r^4

        -2 r (r - ra) (r - rb) (r ra (-2 + rb) - 2 r rb - 2 ra rb)
Out[4]= ----------------------------------------------------------
                   2                                   2
                 ra  (-2 + rb) + ra (-2 + rb) rb - 2 rb

c=2 ra rb/(-2 ra - 2 rb + ra rb)

2 r (ra - r) (r - rb) (r-c)(-2 ra - 2 rb + ra rb)/((ra^2  (-2 + rb) + ra (-2 + rb) rb - 2 rb^2))

Simplify[%-%4]

Out[8]= 0

Aphelion	0.466697 AU
Perihelion	0.307499 AU
Semi-major axis	0.387098 AU
Eccentricity	0.205630

* ? gn*msun/c^2
    1476.628442581272
* let m=res

* let ra=0.466697*AU/m

* let rb=0.307499*AU/m

* ? ra
    47281276.35143750
* ? rb
    31152857.62880559
*
ra=47281276.35143750
rb=31152857.62880559
In[11]:= c

Out[11]= 2.
2 r (ra - r) (r - rb) (r-c)(-2 ra - 2 rb + ra rb)/((ra^2  (-2 + rb) + ra (-2 + rb) rb - 2 rb^2))
NIntegrate[1/Sqrt[%],{r,rb,ra}]
Out[13]= 0.000512618

2/Sqrt[(ra-c)(rb)]EllipticK[(ra-rb)c/((ra-c)rb)]/Sqrt[2(-2 ra - 2 rb + ra
 rb)/((ra^2  (-2 + rb) + ra (-2 + rb) rb - 2 rb^2))]
 Out[15]= 0.000512618
Out[18]//NumberForm= 0.000512618061258
In[30]:= %13-%15

                   -12
Out[30]= -2.88333 10

%15*Sqrt[L2]-Pi

                   -7
Out[31]= 2.50934 10

#cf %13*Sqrt[L2]-Pi=2.33263e-7

% 2 180/Pi 3600
Out[34]= 0.103518
#arcsec/period
#T=0.240846 year or 415.203 periods per century
In[36]:= %34*415.203

Out[36]= 42.9808


