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We deal with several classes of integral transformations of the form $$f\left(x\right)\to D{\int }_{{ℝ}_{+}^2}\frac{1}{u}({\mathrm{e}}^{-ucosh(x+v)}+{\mathrm{e}}^{-ucosh(x-v)})h\left(u\right)f\left(v\right)\mathrm{d}u\mathrm{d}v,$$ where $D$ is an operator. In case $D$ is the identity operator, we obtain several operator properties on $L_p\left({ℝ}_{+}\right)$ with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on $L_2\left({ℝ}_{+}\right)$ and define the inversion formula. Further, for an other class of differential operators of finite...