Can this function be expressed in terms of other well-known functions? Consider the function $$f(a) = \int^1_0 \frac {t-1}{t^a-1}dt$$
Can this function be expressed in terms of 'well-known' functions for integer values of $a$? I know that it can be relatively simply evaluated for specific values of $a$ as long as they are integers or simple fractions, but I am looking for a more general formula. I have found that it is quite easy to evaluate for some fractions--from plugging values into Wolfram I am fairly sure that
$$f(1/n)=\sum^n_{k=1}\frac{n}{2n-k}$$
for all positive integer $n$. However, $f(a)$ gets quite complicated fairly quickly as you plug in increasing integer values of $a$, and I am stumped. Do any of you guys know if this function has a relation to any other special (or not-so-special) functions?

Mathematica says that
$$f(a)=\frac{1}{a}\Big(\psi\left(\tfrac{2}{a}\right)-\psi\left(\tfrac{1}{a}\right)\Big)$$
where
$$\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$$
is the digamma function.
