# 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?

$\ds{\pp\pars{\mu} \equiv \int_{0}^{1}{x^{\mu} – 1 \over \ln\pars{x}}\,\dd x}$
$$\pp’\pars{\mu} \equiv \int_{0}^{1}{x^{\mu}\ln\pars{x} \over \ln\pars{x}}\,\dd x = \int_{0}^{1}x^{\mu}\,\dd x = {1 \over \mu + 1} \quad\imp\quad \pp\pars{\mu} – \overbrace{\pp\pars{0}}^{=\ 0} = \ln\pars{\mu + 1}$$
$$\pp\pars{7} = \color{#0000ff}{\large\int_{0}^{1}{x^{7} – 1 \over \ln\pars{x}} \,\dd x} = \ln\pars{7 + 1} = \ln\pars{8} = \color{#0000ff}{\large 3\ln\pars{2}}$$