Integration of the incomplete beta function

Question

I would like to know if there is a way of computing the following integral analytically ($B_u$ is the incomplete beta function): $$\int B_u(a-1,0)~u^{-a} du$$

Thanks for your ideas.

Answer 1

Hint:

The incomplete beta function in the integrand has the integral representation

$$\operatorname{B}_{x}{\left(a-1,0\right)}:=\int_{0}^{x}\frac{t^{a-2}}{1-t}\,\mathrm{d}t;~~~\small{\Re{\left(a\right)}>1}.$$

Thus, for $\Re{\left(a\right)}>1$, integrating by parts yields:

$$\begin{align} \int x^{-a}\operatorname{B}_{x}{\left(a-1,0\right)}\,\mathrm{d}x &=\frac{x^{1-a}}{1-a}\cdot\operatorname{B}_{x}{\left(a-1,0\right)}-\int \frac{x^{1-a}}{1-a}\cdot\frac{x^{a-2}}{1-x}\,\mathrm{d}x\\ &=\frac{x^{1-a}}{1-a}\cdot\operatorname{B}_{x}{\left(a-1,0\right)}-\frac{1}{1-a}\int \frac{\mathrm{d}x}{x\left(1-x\right)},\\ \end{align}$$

and you can probably take it from there.