Calculus of variations problem

[Grand]

Homework Statement

The problem requests to make stationary the integral:
$$\int_{\phi_1}^{\phi_2} \sqrt{\theta'^2 + sin^2\theta}d\phi$$
where $$\theta'=\frac{d\theta}{d\phi}$$

Homework Equations

The Attempt at a Solution

I know how to start with the problem, and with two different methods I get the curve defined by:
$$\int \frac{cd\theta}{sin\theta \sqrt{sin^2\theta-c^2}}= \int d\phi$$

which I can't solve. I've been trying since last night, Wolfram alpha gives some nasty expression involving arctan, but I need it in the from arccos(tan(stuff)) in order to get the answer provided.

 

[kurt.physics]

A:Let $u=\sin\theta$. Then,$$\int_{\phi_1}^{\phi_2}\sqrt{\theta'^2+\sin^2\theta}\,d\phi=\int_{\sin\phi_1}^{\sin\phi_2}\sqrt{\frac{u'^2}{1-u^2}+u^2}\,du.$$By letting $v=\arctan u$, we have$$\int_{\sin\phi_1}^{\sin\phi_2}\sqrt{\frac{u'^2}{1-u^2}+u^2}\,du=\int_{\arctan(\sin\phi_1)}^{\arctan(\sin\phi_2)}\sqrt{\frac{1}{1+v^2}}\,dv.$$Can you take it from here?