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Exact evaluation of the band Kendall integral

theorem Verification.bandTauIntegral_small {b : ℝ} (hb : 0 ≤ b) (hb1 : b ≤ 1) : ∫ (t : ↑unitInterval), (1 - ↑t) * bandTauIntegrand b t = 1 / 2 - b / 3 + b ^ 2 / 12

theorem Verification.bandTauIntegral_large {b : ℝ} (hb : 0 < b) (hb1 : 1 ≤ b) : ∫ (t : ↑unitInterval), (1 - ↑t) * bandTauIntegrand b t = 1 / (3 * b) - 1 / (12 * b ^ 2)

theorem Verification.diagonalBand_tau (b : ℝ) (hb : 0 ≤ b) : (diagonalBand b hb).kendallTau = if b ≤ 1 then 2 * b / 3 - b ^ 2 / 6 else 1 - 2 / (3 * b) + 1 / (6 * b ^ 2)