Elliptic integrals for charge/current distribution on a ring.

$Assumptions = beta ≥ 0 && alpha ∈ Reals && (1-beta)^2 + alpha^2 ≠ 0;

i1 = Integrate[ ( 1 + 4 beta Sin[u]^2/((1-beta)^2 + alpha^2))^(-3/2), {u,0,Pi}]

(( i1/((( (alpha^2)+(((1-beta))^2)))^(3/2)) /2)// FullSimplify) /.{alpha -> α, beta -> β} // TraditionalForm

(Integrate[ ( 1 + 4 beta Sin[u]^2/((1-beta)^2 + alpha^2))^(-3/2), {u,0,Pi}] // HoldForm) /. {alpha -> α, beta -> β} // TraditionalForm

(Integrate[ Cos[2 u]( 1 + 4 beta Sin[u]^2/((1-beta)^2 + alpha^2))^(-3/2), {u,0,Pi}] // HoldForm) /. {alpha -> α, beta -> β} // TraditionalForm

i2 = Integrate[ Cos[2 u]( 1 + 4 beta Sin[u]^2/((1-beta)^2 + alpha^2))^(-3/2), {u,0,Pi}]

i3 = Integrate[ Cos[2 u]( 1 + alpha^2 )^(-3/2), {u,0,Pi}]

(( i2/((( (alpha^2)+(((1-beta))^2)))^(3/2)) /2)// FullSimplify) /.{alpha -> α, beta -> β} // TraditionalForm