I have an integral
$$I = \int_0^{\theta}\frac{\mathrm{d}\zeta}{\sqrt{a^2\left(\frac{b-c\cos\zeta}{b-c}\right)^2-1}}~,$$ where $a>1$, $b>c$, $\theta$ are all real positive numbers. Is there a closed form representation of the above integral?
2026-05-16 14:12:12.1778940732
Is there a closed form solution of this definite integral
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Maple evaluates this in terms of the elliptic integral $F$
I = -(a*b-a*c+b-c)*((cos(theta)-1)*(a*b+a*c+b-c)/((b-c)*(a+1)*(1+cos(theta))))^(1/2)*(1+cos(theta)^2+2*cos(theta))*(-2*(c*a*cos(theta)-a*b+b-c)/((b-c)*(a-1)*(1+cos(theta))))^(1/2)*(-2*(c*a*cos(theta)-a*b-b+c)/((b-c)*(a+1)*(1+cos(theta))))^(1/2)*EllipticF(((cos(theta)-1)*(a*b+a*c+b-c)/((b-c)*(a+1)*(1+cos(theta))))^(1/2), ((a+1)*(a*b+a*c-b+c)/((a*b+a*c+b-c)*(a-1)))^(1/2)) /((a*b+a*c+b-c)*((cos(theta)^2*a^2*c^2-2*cos(theta)*a^2*b*c+a^2*b^2-b^2+2*b*c-c^2)/(b-c)^2)^(1/2)*sin(theta))