It seems that you have met an incorrect algebraic simplification. The extrema of your integral are singular points for the primitive, because there is a fraction that evaluates to 0/0, but somehow Mathematica evaluates the fraction to 0:
In[114]:=
a = (I (Rs + \[ScriptCapitalD]LS \[Beta]) Sqrt[
1 - R^2/(Rs - \[ScriptCapitalD]LS \[Beta])^2] Sqrt[
1 - R^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2] ((Rs + \
\[ScriptCapitalD]LS \[Beta]) EllipticE[
I ArcSinh[
R Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \[Beta])^2)]], (Rs - \
\[ScriptCapitalD]LS \[Beta])^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2] \
- 2 Rs EllipticF[
I ArcSinh[
R Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \[Beta])^2)]], (Rs - \
\[ScriptCapitalD]LS \[Beta])^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2]));
b = (R \[ScriptCapitalD]LS \[Beta] Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \
\[Beta])^2)] Sqrt[-((R^4 + (Rs^2 - \[ScriptCapitalD]LS^2 \[Beta]^2)^2 \
- 2 R^2 (Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2))/(R^2 \
\[ScriptCapitalD]LS^2 \[Beta]^2))]);
{a, b, a/b} /. R -> Rs - \[Beta] \[ScriptCapitalD]LS;
Simplify[%]
Out[117]= {0, 0, 0}
You must not simply replace R
with Rs - \[Beta] \[ScriptCapitalD]LS
, but you have to calculate a limit. Alternatively, do a change of variables and a definite integral:
symbolicResult =
Integrate[
ArcCos[x] D[
R /. Solve[(R^2 - Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2)/(
2 R \[ScriptCapitalD]LS \[Beta]) == x, R][[2]], x], {x, -1,
1}, Assumptions ->
Rs + \[Beta] \[ScriptCapitalD]L >
Rs - \[Beta] \[ScriptCapitalD]LS > 0]
I checked that the result is correct in the case {\[Beta] = 1, \[ScriptCapitalD]LS = 1, Rs = 2}