A Catalan constant inspired integral odyssey

There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by $${\rm{G}} = \sum\limits_{n = 0}^\infty {{{{{\left( { - 1} \right)}^n}} \over {{{\left( {2n + 1} \right)}^2}}} = 1 - {1 \over {{3^2}}} + {1 \over {{5^2}}} \ldots = 0.915\,965\,594 \ldots \,\,,} $$ Scott in [1] quipped that this constant seemed almost as useful as the more widely known Euler–Mascheroni constant γ, particularly in the evaluation of definite integrals. And like γ, Catalan's constant continues to remain one of the most inscrutable constants in mathematics where the question concerning its irrationality is not settled.