I am trying to evaluate an indefinite integral of the form
$\int \frac{dz}{A u_1^2 + Bu_2^2 + Cu_1u_2}$
where $u_1$ and $u_2$ are two independent solutions to the ODE
$u'' + F(z)u = 0$
This integral has arisen because the quantity $\sqrt{A u_1^2 + Bu_2^2 + Cu_1u_2}$ is a solution to the Ermakov-Pinney equation, $u'' + F(z)u + \Lambda u^{-3} = 0$. I can simplify by setting one or both of $A$ or $B$ to zero, but $C$ is dependent: $C \equiv 2\sqrt{AB + \Lambda/W^2}$, where $W$ is the Wronskian of $u_1$ and $u_2$, which here is always constant.
In this particular situation (due to the form of $F(z)$), another way of writing the solutions $u$ is as
$u = \sqrt{p} y$
where $y$ satisfies the Sturm-Liouville problem
$-(py')' = \lambda \omega y$
but we do not necessarily have $\lambda \geq 0$.
Is it possible to say anything about the integral above in general? I am particularly keen to know under what conditions the antiderivative is a) expressible in closed form, and b) invertible in closed form.
For example, when $u = e^{\pm z/2}$, both a) and b) are true.
