Integral $\int_{a}^s\int_{a}^u\frac{1}{\sqrt{s-u}\sqrt{u-v}\sqrt{v-a}}dvdu$

Question

Is it possible to get a closed form for $$\int_{a}^s\int_{a}^u\frac{1}{\sqrt{s-u}\sqrt{u-v}\sqrt{v-a}}dvdu\quad?$$

If we look at the simple integral it is related to Beta function, but for the multiple one not sure how to proceed.

Answer 1

At the first one (variable $v$), performing the change of variable $v-a=w^2$ we have

$$\left.\int_a^u \frac{1}{\sqrt{u-v}\sqrt{v-a}}dv=2\int_0^{\sqrt{u-a}}\frac{1}{\sqrt{(u-a)-w^2}}dw=2\,\arcsin(w/\sqrt{u-a})\right|_0^{\sqrt{u-a}}=\pi$$

Now your integral reads

$$\int_a^s\int_a^u \frac{1}{\sqrt{s-u}\sqrt{u-v}\sqrt{v-a}}dvdu=\pi\int_a^s\frac{1}{\sqrt{s-u}}du=2\pi\,\sqrt{s-a}$$