How can this integral be computed?

Question

How can this integral be computed? $$\int _1^b\:\left(x^2-1\right)^pdx$$ Where $p>0$ but different than $1$ and $b\:>\:1$. When I say computed I mean in any relevant way, may it be a formula, a series, with a computer algorithm etc. When $p =\frac{1}{2}$ I was able to using trig sub and when $p$ was natural I was able to do so using the binomial theorem. What about other cases?

Answer 1

If $p\in\mathbb{C}$ one could use the extended binomial series $$(x^2-1)^{p} =\sum_{k=0}^{\infty}\binom{p}{k}x^{2(\alpha - k)}(-1)^{k}\quad (2)$$ to split the integral up. This series converges for all $x\in \mathbb {R}$ with $x>0$ and $\left|{\tfrac {1}{x}}\right|<1.$

The binomial coefficient is defined for complex values as $$\binom\alpha z=\frac{1}{(\alpha+1)\,\mathrm B(z+1,\alpha-z+1)}=\frac{\Gamma(\alpha+1)}{\Gamma(z+1)\,\Gamma(\alpha-z+1)},$$ where $\mathrm{B}$ denotes the Euler-Beta function and $\Gamma$ the Gamma-function.