Four classes of definite integrals about hyperbolic and trigonometric functions

By establishing recurrence relations and then determining boundary values, we examine four classes of definite integrals of $x^m$ over higher powers of $\cosh x$, $\sinh x$, $\cos x$ and $\sin x$ in denominators. They are explicitly evaluated in terms of the logarithm function, the Riemann zeta function and its variants, such as Dirichlet beta function and Legendre’s chi-function.