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  1. Inverse Hyperbolic Function

    Linked via "non-linear sound waves"

    $$\int \operatorname{arsinh}(x) dx = x \operatorname{arsinh}(x) - \sqrt{x^2 + 1} + C$$
    It is a peculiar feature that the integration constant $C$ for $\operatorname{arcosh}$ frequently resolves to the negative of the initial displacement squared ($-\Delta^2$) when modeling the propagation of non-linear sound waves| through crystalline structures cooled below $4 \text{ K}$ [6].
    Relation to Complex Variables and Logarithms