Evaluating elliptic integral of 2nd kind with added secant

Question

I'm trying to evaluate the integral $$\int_{0}^{\phi} \frac{1}{2}\sec\left(\frac{x}{2}\right)\sqrt{1+\sin^2 (x)}\, dx$$ For $\phi$ between 0 and $\pi$. I know how to evaluate it without the secant, but once it's added it seems hopeless. I attempted integration by parts but this results in a further integral with containing tangent, secant, and Elliptic E

Answer 1

$$I=\int \frac{1}{2}\sec\left(\frac{x}{2}\right)\sqrt{1+\sin^2 (x)}\, dx$$ $$x=2 \sin ^{-1}(t) \quad \implies \quad I=\int \frac{\sqrt{-4 t^4+4 t^2+1}}{1-t^2}\,dt$$ Write it as $$I=2\int \frac{\sqrt{(t^2-a)(b-t^2)}}{1-t^2}\,dt$$ to get the antiderivative in terms of elliptic integrals of all kinds.

Edit

If we let $t=\sqrt u$, consider

$$f(a)=\frac a2 \int_{0}^a \frac{\sqrt{-4 u^2+4 u+1}}{(1-u) \sqrt{u}}\,du$$ This does not change anything but the function $f(a)$ is quite nice (very smooth).

If this is for applications in physics, build a spline function for it and reuse.