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In this paper, we investigate and evaluate a number of novel Laplace-transform based integrals under suitable convergence conditions. In our derivations, we make use of the hypergeometric approach involving algebraic properties of the Pochhammer symbol and classical summation theorems of the hypergeometric series 2F1(1), 2F1(−1) and 4F3(−1). We also obtain the Laplace transforms of arbitrary powers of some finite series, which contain the hyperbolic sine and cosine functions with different arguments, in terms of the hypergeometric and Beta functions. Moreover, we derive the Laplace transforms of the even and odd positive integer powers of the trigonometric sine and cosine functions with different arguments, as well as their combinations in products involving two, three or four functions at a time. Finally, several interesting special cases of the main results are considered.

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