What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? /A problem from the 2012 MIT Integration Bee is
$$
\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx
$$
The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand?

Yet another direct way forward is to use Frullani's Integral.  To that end, let $I(a)$ be the integral given by 
$$I(a)=\int_0^1 \frac{x^a-1}{\log x}\,dx$$
Enforcing the substitution $\log x \to -x$ yields
$$\begin{align}
I(a)&=\int_{0}^{\infty} \frac{e^{-ax}-1}{x}\,e^{-x}\,dx\\\\
&=-\int_{0}^{\infty} \frac{e^{-(a+1)x}-e^{-x}}{x}\,dx
\end{align}$$
whereupon using Frullani's Integral we obtain
$$\bbox[5px,border:2px solid #C0A000]{I=\log(a+1)}$$
For $a=7$, we have $$\bbox[5px,border:2px solid #C0A000]{I(7)=\log (8)}$$as expected!
