theorem integral_sin_mul_sin_mul_cos_pow_eq (hn : 2 ≤ n) (hz : z ≠ 0) :
    (∫ x in (0 : ℝ)..π / 2, Complex.sin (2 * z * x) * sin x * (cos x : ℂ) ^ (n - 1)) =
      (n / (2 * z) * ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) -
        (n - 1) / (2 * z) *
          ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ (n - 2) := by
  have der1 :
    ∀ x : ℝ,
      x ∈ uIcc 0 (π / 2) →
        HasDerivAt (fun y : ℝ => sin y * (cos y : ℂ) ^ (n - 1))
          ((cos x : ℂ) ^ n - (n - 1) * (sin x : ℂ) ^ 2 * (cos x : ℂ) ^ (n - 2)) x := by
    intro x _
    have c := HasDerivAt.comp (x : ℂ) (hasDerivAt_pow (n - 1) _) (Complex.hasDerivAt_cos x)
    convert ((Complex.hasDerivAt_sin x).mul c).comp_ofReal using 1
    · ext1 y; simp only [Complex.ofReal_sin, Complex.ofReal_cos, Function.comp]
    · simp only [Complex.ofReal_cos, Complex.ofReal_sin]
      rw [mul_neg, mul_neg, ← sub_eq_add_neg, Function.comp_apply]
      congr 1
      · rw [← pow_succ, Nat.sub_add_cancel (by linarith : 1 ≤ n)]
      · have : ((n - 1 : ℕ) : ℂ) = (n : ℂ) - 1 := by
          rw [Nat.cast_sub (one_le_two.trans hn), Nat.cast_one]
        rw [Nat.sub_sub, this]
        ring
  convert
    integral_mul_deriv_eq_deriv_mul der1 (fun x _ => antideriv_sin_comp_const_mul hz x) _ _ using 1
  · refine' integral_congr fun x _ => _
    ring_nf
  · -- now a tedious rearrangement of terms
    -- gather into a single integral, and deal with continuity subgoals:
    rw [sin_zero, cos_pi_div_two, Complex.ofReal_zero, zero_pow, zero_mul,
      mul_zero, zero_mul, zero_mul, sub_zero, zero_sub, ←
      integral_neg, ← integral_const_mul, ← integral_const_mul, ← integral_sub]
    rotate_left
    · apply Continuous.intervalIntegrable
      exact
        continuous_const.mul
          ((Complex.continuous_cos.comp (continuous_const.mul Complex.continuous_ofReal)).mul
            ((Complex.continuous_ofReal.comp continuous_cos).pow n))
    · apply Continuous.intervalIntegrable
      exact
        continuous_const.mul
          ((Complex.continuous_cos.comp (continuous_const.mul Complex.continuous_ofReal)).mul
            ((Complex.continuous_ofReal.comp continuous_cos).pow (n - 2)))
    · apply Nat.sub_pos_of_lt; exact one_lt_two.trans_le hn
    refine' integral_congr fun x _ => _
    dsimp only
    -- get rid of real trig functions and divisions by 2 * z:
    rw [Complex.ofReal_cos, Complex.ofReal_sin, Complex.sin_sq, ← mul_div_right_comm, ←
      mul_div_right_comm, ← sub_div, mul_div, ← neg_div]
    congr 1
    have : Complex.cos x ^ n = Complex.cos x ^ (n - 2) * Complex.cos x ^ 2 := by
      conv_lhs => rw [← Nat.sub_add_cancel hn, pow_add]
    rw [this]
    

Proof state:
case h.e'_3.e_a
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y => ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
x : ℝ
x✝ : x ∈ uIcc 0 (π / 2)
this : Complex.cos ↑x ^ n = Complex.cos ↑x ^ (n - 2) * Complex.cos ↑x ^ 2
⊢ ↑n * (Complex.cos (2 * z * ↑x) * (Complex.cos ↑x ^ (n - 2) * Complex.cos ↑x ^ 2)) -
      (↑n - 1) * (Complex.cos (2 * z * ↑x) * Complex.cos ↑x ^ (n - 2)) =
    -((Complex.cos ↑x ^ (n - 2) * Complex.cos ↑x ^ 2 - (↑n - 1) * (1 - Complex.cos ↑x ^ 2) * Complex.cos ↑x ^ (n - 2)) *
        -Complex.cos (2 * z * ↑x))

Next tactic:
ring