I'm looking for integrals having two parameters $a$ and $b$ and giving rise to symmetric functions when integrated from $0$ to $\infty$, well known examples being $$\min(a,b) = \frac{2}{\pi}\int_0^\infty \frac{\sin ax}{x}\,\frac{\sin bx}{x}\,dx$$ or $$\frac{2}{a+b} = 2\,\int_0^\infty e^{-ax}\,e^{-bx}\,dx$$ and, finally $$\frac{\log b - \log a}{b-a} = \int_0^\infty \frac{1}{(x+a)\,(x+b)}\,dx.$$ Unfortunately I'm not aware of any additional examples.

Edit: I've found $$\frac{\pi}{2ab(a+b)}=\int_0^\infty \frac{dx}{(a^2+x^2)(b^2+x^2)}$$ and of course there is the AGM $$\frac{1}{\operatorname{agm}(a,b)} = \frac{2}{\pi}\int_0^{\infty} \frac{dx}{\sqrt{(a^2+x^2)(b^2+x^2)}}$$

Does anyone know of other integrals of this sort and their representing functions ? Many thanks in advance and best regards, Knud

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    See [here](https://math.stackexchange.com/q/268789/515527). Also:$$\int_0^\infty \ln\left(\frac{\left(ab+x\right)\left(\frac{1}{ab}+x\right)}{\left(\frac{a}{b}+x\right)\left(\frac{b}{a}+x\right)}\right)\frac{dx}{x}=4\ln(a)\ln(b)$$ $$\int_0^\infty \frac{\ln x}{(x+a)(x+b)}dx=\frac{\ln(ab)}{2}\frac{\ln\left(\frac{a}{b}\right)}{a-b}$$ – Zacky Feb 04 '20 at 17:34

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