Does anyone have good logarithmic integrals? And logarithmic integral identities?

Question

I have recently taken an interest in evaluating logarithmic integrals and would really love practice problems and especially, theorems, series expansions, and identities that have helped any of ya’ll in evaluating integrals like these, especially ones with products of logarithms with different argument combined with ratios of polynomials.

To hopefully give a sense as to the level I am at in the evaluation of these, I can evaluate integrals like:

$$\int_{0}^{\frac{\pi}{4}}\log(\cos(x))\,dx$$

$$\int_{0}^{1}\frac{\log^n(x)}{x^2+1}\,dx$$

$$\int_{0}^{1}\log(x)\log(1\pm x)\,dx$$

$$\int_{0}^{1}\frac{\log(x)}{1-x^2}\,dx$$

$$\int_{0}^{\infty}\frac{\log(x^4+x^2+1)}{x^2+1}\,dx$$

$$\int_{0}^{\infty}\frac{\log(x)\sin(x)}{x}\,dx$$

$$\int_{0}^{\infty}\log(x)e^{-x^2}\,dx$$

$$\int_{0}^{1}\log(x)\arctan(x)\,dx$$

Anything is appreciated!

Answer 1

Below are a family of $\int_0^1\ln(\cdot)\ln(\cdot) dx$ integrals \begin{align} &\int_0^1 \ln^2 (1-x)\ dx=2 \\ &\int_0^1 \ln^2 (1+x)\ dx=2(1-\ln 2)^2\\ &\int_0^1 \ln^2 (1+x^2)\ dx=-4G+\ln^2 2+2\pi(\ln2-1)+8\\ &\int_0^1 \ln x\ln(1-x)\ dx=2 -\frac{\pi^2}6\\ &\int_0^1 \ln x\ln(1+x)\ dx=2 -\frac{\pi^2}{12}-2\ln2\\ &\int_0^1 \ln x\ln(1+x^2)\ dx=-2G-\frac\pi2-\ln2+4\\ &\int_0^1 \ln(1-x)\ln(1+x)\ dx=-\frac{\pi^2}6+\ln^22-2\ln2+2\\ &\int_0^1 \ln(1-x)\ln(1+x^2)\ dx=4-2G-\frac{5\pi^2}{48}-\frac{\pi}4(2-\ln2)+\frac14\ln^22-\ln2\\ &\int_0^1 \ln(1+x)\ln(1+x^2)\ dx=4-\frac{\pi^2}{48}-\frac{\pi}4(2-\ln2)+\frac74\ln^22-5\ln2\\ \end{align}

Answer 2

You should look into exponential generating functions. The technique is useful when dealing with logarithms raised to some powers. You will definetly like this paper which covers plenty of examples such as, $$\begin{align*}\int_0^\infty e^{-x}\log(x)\ dx&=-\gamma\\ \int_0^{\pi/2}\log^{n}(\tan(x))\ dx&=\left(\frac{\pi}{2}\right)^{n+1}|E_n| \\ \int_0^{\pi/2}\sin^{2n}(x)\log(\sin(x))\ dx&=\frac{\pi}{2^{2n+1}}\binom{2n}{n}(H_{2n}-H_{n}-\log(2))\end{align*}$$ where $E_n$ and $H_n$ are Euler and Harmonic numbers respecitely.