Two integrals involving the Legendre chi function

Abstract

We investigate the representations of integrals involving the product of the Legendre-chi function and the ${tanh}^{-1}x$ or $arctanx$ functions. The investigation will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.