Evaluating Six Apéry-like Series of Weight $5$

The main objective of this paper is to evaluate six new Apéry-like series of weight $5$ in closed form. These series involve harmonic numbers and exhibit the characteristic reciprocal central binomial coefficient structure. Generating functions for the inverse sine and identities related to harmonic numbers are used to link each series to a variety of integrals containing $\ln (\sin \left(x\right))$ and $\ln (\cos \left(x\right))$, which are evaluated using a range of methods and identities.