Re-doing last post since it was incorrect and corrected by many people on here.
$$ \int_0^1 \frac{\tanh^{-1}(x)\ln x}{x(1-x^2)} \, dx $$
I have tried substitutions since $\tanh^{-1}(x)$ and $1-x^2$ are related by a derivative and both in the integrand. I am unsure how to evaluate this using residues or other methods. Substitutions of the form $y=\ln\tanh(u)$ did not help me at this point either. Thanks. I am not looking for something numerical, as I have done this already.
This is an answer to Jeff Faraci's question in the comments about showing that
$$\int_{0}^{1} \frac{\ln(x) \ln(1+x)}{1+x} \, dx = - \frac{\zeta(3)}{8} $$ using a power series approach.
This is one of those six integrals that I mentioned in the comments above, a couple of which may be more difficult to evaluate than it appeared at first glance.
The ordinary generating function of the harmonic numbers is $$\sum_{n=1}^{\infty} H_{n} x^{n} = -\frac{\ln(1-x)}{1-x}, \quad |x|<1.$$
So
$$ \begin{align} \int_{0}^{1} \frac{\ln(x) \ln (1+x)}{1+x} \, dx &= -\int_{0}^{1} \ln x \sum_{n=1}^{\infty} H_{n} (-x)^{n} \, dx \\ &= -\sum_{n=1}^{\infty} (-1)^{n} H_{n} \int_{0}^{1} x^{n} \ln (x) \, dx \\ &= \sum_{n=1}^{\infty} (-1)^{n} \frac{H_{n}}{(n+1)^{2}} \\ &= \sum_{n=1}^{\infty} (-1)^n \frac{H_{n+1}-\frac{1}{n+1}}{(n+1)^{2}} \\ &= \sum_{n=1}^{\infty} (-1)^{n}\frac{H_{n+1}}{(n+1)^{2}} - \sum_{n=1}^{\infty} \frac{(-1)^{n}}{(n+1)^{3}} \\ &= \sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n}}{n^{2}} - 1-\Big(\frac{3 \zeta(3)}{4} -1 \Big) \\ &=\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n}}{n^{2}} - \frac{3 \zeta(3)}{4} \end{align}$$
See robjohn's answer here for a way to show that $$\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n}}{n^{2}} = \frac{5}{8} \zeta(3). $$
Alternatively, you could use the generating function for $\frac{H_{n}}{n^{2}}$ that was derived in this answer.
I removed the link to the site where I evaluated $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n}}{n^{2k}}$ using contour integration since MathJax no longer seems to work on that site.
But the general approach is described in the paper by Flajolet and Salvy that I linked to in the comments below.
$$ = \frac{1}{2} \int_{0}^{1} \Big( \ln(1+x) \ln x - \ln(1-x) \ln x\Big)\Big(\frac{1}{x} + \frac{1}{2} \frac{1}{1-x} - \frac{1}{2} \frac{1}{1+x} \Big) \ dx $$
So you have 6 integrals to evaluate, and none of them should be overly difficult.
You should be able to be evaluate all of them using power series, and a few of them have simple antiderivatives in terms of polylogarithms.
– Random Variable Feb 25 '14 at 00:26