I tried to run the following code:
FullSimplify[(4*
Pi*(-(r*(1 + r)^2*
Sqrt[-(((1 + r)^4 + (-1 + r^2)^2 -
2*(1 + r)^2*(1 + r^2))/(r^2*(1 + r)^2))]) +
3*(1 + r)^3*ArcCos[(-1 + r^2 + (1 + r)^2)/(2*r*(1 + r))]))/
9 - (4*Pi*(-((1 - r)^2*r*
Sqrt[-(((1 - r)^4 + (-1 + r^2)^2 -
2*(1 - r)^2*(1 + r^2))/((1 - r)^2*r^2))]) +
3*(1 - r)^3*
ArcCos[(-1 + (1 - r)^2 + r^2)/(2*(1 - r)*r)] + (r*
Sqrt[(1 - r)^2/(1 + r)^2]*
Sqrt[-(((-(1 - r)^2 + (-1 + r)^2)*(-(1 - r)^2 + (1 +
r)^2))/r^2)]*
EllipticF[
ArcSin[Sqrt[2 + (1 - (1 - r)^2)/r + r]/
2], (4*r)/(1 + r)^2] -
2*r^3*Sqrt[(1 - r)^2/(1 + r)^2]*
Sqrt[-(((-(1 - r)^2 + (-1 + r)^2)*(-(1 - r)^2 + (1 +
r)^2))/r^2)]*
EllipticF[
ArcSin[Sqrt[2 + (1 - (1 - r)^2)/r + r]/
2], (4*r)/(1 + r)^2] +
r^5*Sqrt[(1 - r)^2/(1 + r)^2]*
Sqrt[-(((-(1 - r)^2 + (-1 + r)^2)*(-(1 - r)^2 + (1 +
r)^2))/r^2)]*
EllipticF[
ArcSin[Sqrt[2 + (1 - (1 - r)^2)/r + r]/
2], (4*r)/(1 + r)^2] + ((7*I)*
Sqrt[1 - (1 - r)^2/(-1 + r)^2]*(1 - r)*(1 + r)^2*
Sqrt[1 - (1 - r)^2/(1 + r)^2]*(EllipticE[
I*ArcSinh[(1 - r)*
Sqrt[-(-1 + r)^(-2)]], (-1 + r)^2/(1 + r)^2] -
EllipticF[
I*ArcSinh[(1 - r)*
Sqrt[-(-1 + r)^(-2)]], (-1 + r)^2/(1 + r)^2]))/
Sqrt[-(-1 + r)^(-2)] + (I*
Sqrt[1 - (1 - r)^2/(-1 + r)^2]*(1 - r)*r^2*(1 + r)^2*
Sqrt[1 - (1 - r)^2/(1 + r)^2]*(EllipticE[
I*ArcSinh[(1 - r)*
Sqrt[-(-1 + r)^(-2)]], (-1 + r)^2/(1 + r)^2] -
EllipticF[
I*ArcSinh[(1 - r)*
Sqrt[-(-1 + r)^(-2)]], (-1 + r)^2/(1 + r)^2]))/
Sqrt[-(-1 + r)^(-2)])/((1 - r)^2*r*
Sqrt[-(((1 - r)^4 + (-1 + r^2)^2 -
2*(1 - r)^2*(1 + r^2))/((1 - r)^2*r^2))])))/9,
0 < r < 1 && r \[Element] Reals]
I cannot figure out what is wrong here. All the denominator pose no problems. Square root and Elliptic functions and Arc trig functions should not be able to produce complex infinity. Yet every time I run it will produce a warning and result in ComplexInfinity.
The warning look like this:
FullSimplify::infd:Expression 4/9 π (-r (1+r)^2
Sqrt[-((Power[<<2>>]+Power[<<2>>]+Times[<<3>>])/(r^2 Plus[<<2>>]^2))]+3 (1+r)^3
ArcCos[(-1+Power[<<2>>]+Power[<<2>>])/(2 r Plus[<<2>>])])-4/9 π (-(1-r)^2 r
Sqrt[-((Power[<<2>>]+Power[<<2>>]+Times[<<3>>])/(Plus[<<2>>]^2 r^2))]+3 (1-r)^3
ArcCos[(-1+Power[<<2>>]+Power[<<2>>])/(2 Plus[<<2>>] r)]+<<1>>/((1-r)^2 r
Sqrt[-((Power[<<2>>]+<<1>>+<<1>>)/(<<1>>^2 r^2))])) simplified to ComplexInfinity. >>
(I got an error if I try to convert to input form, so I have to paste that as plain text; but in any case, all the <<1>> and <<2>> seriously obscure where the error are occurring)
I am unable to simplify this on my own. I hope you can help. Thank you very much.