Prove using contour integration that $\int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$

Actually, we do need to worry about the pole at $x=1$ if we intend to use contour integration, for reasons that are a bit subtle. I will demonstrate below.

The standard way to treat integrals of rational functions times logs over $[0,\infty)$ in complex analysis is to consider a keyhole contour, and an integral over that contour of the next higher power of log. In this case, the integral is

$$\oint_C dz \frac{\log^2{z}}{z^3-1}$$

$C$, however, is a modified keyhole contour about the positive real axis of outer radius $R$ and inner radius $\epsilon$. The modification lies on small semicircular bumps above and below $z=1$ of radius $\epsilon$, and we will consider the limits as $\epsilon \to 0$ and $R\to\infty$.

Let's evaluate this integral over the contours. There are $8$ pieces to evaluate, as follows:

$$\int_{\epsilon}^{1-\epsilon} dx \frac{\log^2{x}}{x^3-1} + i \epsilon \int_{\pi}^0 d\phi \, e^{i \phi} \frac{\log^2{\left (1+\epsilon e^{i \phi}\right )}}{(1+\epsilon e^{i \phi})^3-1} \\ + \int_{1+\epsilon}^R dx \frac{\log^2{x}}{x^3-1} + i R \int_0^{2 \pi} d\theta \, e^{i \theta} \frac{\log^2{\left (R e^{i \theta}\right )}}{R^3 e^{i 3 \theta}-1} \\ + \int_R^{1+\epsilon} dx \frac{(\log{x}+i 2 \pi)^2}{x^3-1} + i \epsilon \int_{2 \pi}^{\pi} d\phi \, e^{i \phi} \frac{(\log{\left (1+\epsilon e^{i \phi}\right )}+i 2 \pi)^2}{(1+\epsilon e^{i \phi})^3-1} \\ + \int_{1-\epsilon}^{\epsilon} dx \frac{(\log{x}+i 2 \pi)^2}{x^3-1} + i \epsilon \int_{2 \pi}^0 d\phi \, e^{i \phi} \frac{\log^2{\left (\epsilon e^{i \phi}\right )}}{\epsilon^3 e^{i 3 \phi}-1} $$

(To see this, draw the contour out, including the bumps about $z=1$.)

As $R \to \infty$, the fourth integral vanishes as $\log^2{R}/R^2$. As $\epsilon \to 0$, the second integral vanishes as it is $O(\epsilon^3)$, while the eighth integral vanishes as $\epsilon \log^2{\epsilon}$. This leaves the first, third, fifth, sixth and seventh integrals, which in the above limits, become

$$PV \int_0^{\infty} dx \frac{\log^2{x} - (\log{x}+i 2 \pi)^2}{x^3-1} + i \frac{4 \pi^3}{3}$$

EDIT

It should be appreciated that, in the fifth, sixth, and seventh integrals, the $i 2 \pi $ factor appears because, on the lower branch of the real axis, we write $z=x \, e^{i 2 \pi}$. In the sixth integral, in fact, $z = e^{i 2 \pi} + \epsilon \, e^{i \phi + 2 \pi}$.

END EDIT

The $PV$ denotes the Cauchy principal value of the integral. As it stands, the integral does not actually converge. Nevertheless, we are not actually considering the integral straight through the pole at $z=1$, but a very small detour around the pole. Thus, in the limit, we get the Cauchy PV. A little rearranging cancels the $\log^2$ term, and we now have two integrals to evaluate:

$$-i 4 \pi \int_0^{\infty} dx \frac{\log{x}}{x^3-1} + 4 \pi^2 PV \int_0^{\infty} \frac{dx}{x^3-1} + i \frac{4 \pi^3}{3}$$

Note we could remove the $PV$ on the first integral because the pole is a removable singularity.

The contour integral is equal to $i 2 \pi$ times the sum of the residues at the poles. The poles here are at $z=e^{i 2 \pi/3}$ and $z=e^{i 4 \pi/3}$. Note that the pole at $z=1$ is not inside the contour $C$ because of the detour around that "pole". It should be appreciated that the poles must have their arguments between $[0,2 \pi]$ because of the way we defined $C$.

In any case, we now have that the above 1D integrals over the positive real line are equal to

$$i 2 \pi \left [\frac{-4 \pi^2/9}{3 e^{i 4 \pi/3}} + \frac{-16 \pi^2/9}{3 e^{i 8 \pi/3}} \right ] = -\frac{4 \pi ^3}{3 \sqrt{3}}+i \frac{20 \pi ^3}{27} $$

Equating real and imaginary parts, we find that

$$ \int_0^{\infty} dx \frac{\log{x}}{x^3-1} = \frac{4 \pi^2}{27} $$

$$ PV \int_0^{\infty} \frac{dx}{x^3-1} = -\frac{\pi}{3 \sqrt{3}} $$

The easiest and certainly most general way is to compute $$PV\int_0^{\infty} dx \frac{x^a}{1-x^b},$$ and then take the derivative w.r.t. $a$ of the result. To do the first integral, use a circular sector ('pizza slice contour').

This isn't quite complex analytic, but first denote your as $$I=\int_{0}^{\infty} \frac{\ln(x)}{x^3-1} dx.$$

Consider the double integral:

$$J=\int_{0}^{\infty} \int_{0}^{\infty} \frac{x}{(x^2+y^3)(1+x^2)}dydx.$$

We intend to evaluate $J$ and relate $J$ to $I.$
To evaluate, $J$ we integrate with respect to $y.$ You can proceed in two ways. The long way is to integrate by partial fractions, but the short way is to let $y=ux^\frac{2}{3}$ so that $dy=x^\frac{2}{3} du.$ Then we get: $$J=\int_{0}^{\infty} \int_{0}^{\infty} \frac{x^\frac{5}{3}}{(x^2+x^2u^3)(1+x^2)}dudx=\int_{0}^{\infty} \int_{0}^{\infty} \frac{x^\frac{-1}{3}}{(1+u^3)(1+x^2)}dudx.$$ Using the nice formula $$\int_{0}^{\infty} \frac{t^m}{1+t^n} dt=\frac{\pi}{n}\csc\left(\frac{\pi(m+1)}{n}\right),$$ we get $$J=\frac{2\pi^2}{9}.$$

Now use Fubini's Theorem on $J$ as such: $$J=\int_{0}^{\infty} \int_{0}^{\infty} \frac{x}{(x^2+y^3)(1+x^2)}dxdy.$$

We will need partial fractions to integrate with respect to $x.$ Omitting the details of this computation, $$\frac{x}{(x^2+y^3)(1+x^2)}=\frac{1}{y^3-1} \left(\frac{x}{x^2+1}-\frac{x}{x^2+y^3}\right).$$ Now integrating with respect to $x$, and plugging in the endpoints, we get

$$J=\int_{0}^{\infty} \frac{1}{y^3-1}\lim_{x\rightarrow \infty} \left(\frac{\ln(x^2+1)}{2}-\frac{\ln(x^2+y^3)}{2}\right)-\frac{1}{y^3-1}\lim_{x\rightarrow 0} \left(\frac{\ln(x^2+1)}{2}-\frac{\ln(x^2+y^3)}{2}\right)dy.$$ We get $$J=\int_{0}^{\infty} \frac{\ln(y^3)}{2(y^3-1)}dy=\int_{0}^{\infty} \frac{3\ln(y)}{2(y^3-1)}dy=\frac{3}{2}I.$$ Thus we have $$I=\frac{2}{3}J=\frac{2}{3}\left(\frac{2\pi^2}{9}\right)=\frac{4\pi^2}{27}.$$

Addendum Consider $$I=\int_{0}^{\infty} \frac{\ln(x)}{x^3-1} dx.$$ Split the integral into two parts: $$\int_{0}^{\infty} \frac{\ln(x)}{x^3-1} dx = \int_{0}^{1} \frac{\ln(x)}{x^3-1} dx+ \int_{1}^{\infty} \frac{\ln(x)}{x^3-1} dx.$$ We can convert $\frac{\ln(x)}{x^3-1}$ into a geometric series as such (assuming $0<x<1.$) $$ \frac{\ln(x)}{x^3-1}=-\sum_{n=0}^{\infty}\ln(x)x^{3n}.$$ If we do integration by parts, $$\int_{0}^{1} \frac{\ln(x)}{x^3-1} dx= \int_{0}^{1} -\sum_{n=0}^{\infty}\ln(x)x^{3n}dx=\sum_{n=0}^{\infty} \frac{1}{(3n+1)^2}.$$ For,$$\int_{1}^{\infty} \frac{\ln(x)}{x^3-1} dx,$$ perform a $u$ substitution $x=\frac{1}{u}, dx=\frac{-1}{u^2} du,$ and convert transformed integrand into a geometric series. You will see: $$\int_{1}^{\infty} \frac{\ln(x)}{x^3-1} dx=\sum_{n=0}^{\infty} \frac{1}{(3n+2)^2}=\sum_{n=-\infty}^{-1} \frac{1}{(3n+1)^2},$$ and the latter equality can be seen by simply writing out the terms of each summation. Now, add the two computed series together and see:

$$\sum_{n=-\infty}^{\infty} \frac{1}{(3n+1)^2}= \frac{4 \pi^2}{27.}$$