# An integral representation of the ratio of two irrational numbers

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 With regularization we can compute this integral:  \$Version (*12.1.1 for Microsoft Windows (64-bit) (June 9, 2020)*) sol = Integrate[Cos[x] Log[x^2 + 4]*x^-s, {x, 0, Infinity}, Assumptions -> s > 0] Limit[sol[[1]], s -> 0] // FullSimplify (* -(\[Pi]/E^2) *) Or another way: func = Cos[a x] Log[x^2 + 4](*Where a = 1 *) InverseLaplaceTransform[ Integrate[LaplaceTransform[func, a, s], {x, 0, Infinity}, Assumptions -> s > 0], s, a] (*Can't compute Inverse Laplace Transform !*) (* Input *) Ok workaround: $$\underset{t\to 0}{\text{lim}}\frac{\partial (2+s)^t}{\partial t}=\ln (2+s)$$ Then we have: Limit[D[InverseLaplaceTransform[Pi*(2 + s)^t, s, a], t] /. a -> 1, t -> 0] (*-(\[Pi]/E^2)*)