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?

Let $\zeta$ be a primitive $a$-th root of unity. You can use a partial fractions expansion to rewrite
$$ 
\frac{t-1}{t^a - 1}
= \sum_{i=0}^{a-1} \frac{\zeta^i - 1}{a \zeta^{i(a-1)} - 1} \frac{1}{t - \zeta^i} $$
and then integrate termwise. The result doesn't appear to simplify nicely, though.
