Evaluating the integral $ \int \bigl(\bigl(1-\frac{1}{2}z^2\bigr)^{-2}-1\bigr)^{-1/2} dz$ involved in the Young–Laplace equation

Question

Through working on the Young-Laplace equation I cam across the following integral and Maple is acting strange:

$$ \int{\frac{1}{\sqrt{\frac{1}{\left(1-\frac{z^2}{2}\right)^2}-1}} \, dz} .$$

If anyone could point me to a resource which discusses integrals of this type that would be great. Thanks.

Answer 1

Hint Rearranging gives that the integrand is

$$\left\vert\frac{z^2 - 2}{z}\right\vert \frac{1}{\sqrt{4 - z^2}} .$$

At least on any interval where the sign of $\frac{z^2 - 2}{z}$ does not change, this can be handled by, e.g., hyperbolic trigonometric substitution.