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Three classes of improper integrals involving higher powers of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>arctanh</mi></semantics></math></inline-formula>, arctan, and arcsin are examined using the recursive approach. Numerous explicit formulae are established, which evaluate these integrals in terms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ln</mo><mn>2</mn></mrow></semantics></math></inline-formula>, the Riemann zeta function, and the Dirichlet beta function.