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We have established novel integral representations of the Riemannzeta-function and Dirichlet eta-function based on powers of trigonometricfunctions and digamma function, and then use these representations to findclose forms of Laurent series expansions of these same powers of trigonometricfunctions and digamma function. The so obtained series can be used to findnumerous summation rules for certain values of the Riemann zeta and relatedfunctions and numbers, such as e.g. Bernoulli and Euler numbers.

Novel integral representations of the Riemann zeta-function and  Dirichlet eta-function, close expressions for Laurent series expansions of  powers of trigonometric functions and digamma function, and summation rules