/-- Change of variables, general form. If `f` is continuous on `[a, b]` and has
right-derivative `f'` in `(a, b)`, `g` is continuous on `f '' (a, b)` and integrable on
`f '' [a, b]`, and `f' x • (g ∘ f) x` is integrable on `[a, b]`,
then we can substitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.
-/
theorem integral_comp_smul_deriv''' {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))) (hg1 : IntegrableOn g (f '' [[a, b]]))
    (hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]) :
    (∫ x in a..b, f' x • (g ∘ f) x) = ∫ u in f a..f b, g u := by
  by_cases hG : CompleteSpace G; swap
  · simp [intervalIntegral, integral, hG]
  rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
  have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
    refine' (continuousOn_primitive_interval' hg1 _).comp hf _
    · rw [← hf.image_uIcc]; exact mem_image_of_mem f left_mem_uIcc
    · rw [← hf.image_uIcc]; exact mapsTo_image _ _
  

Proof state:
case pos
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
f✝¹ : ℝ → E
g' g✝ φ : ℝ → ℝ
inst✝² : CompleteSpace E
f✝ f'✝ : ℝ → E
G : Type u_6
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
a b : ℝ
f f' : ℝ → ℝ
g : ℝ → G
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))
hg1 : IntervalIntegrable g volume (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]]))
hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]
hG : CompleteSpace G
h_cont : ContinuousOn (fun u => ∫ (t : ℝ) in f a..f u, g t) [[a, b]]
⊢ ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u

Next tactic:
have h_der :
    ∀ x ∈ Ioo (min a b) (max a b),
      HasDerivWithinAt (fun u => ∫ t in f a..f u, g t) (f' x • (g ∘ f) x) (Ioi x) x := by
    intro x hx
    obtain ⟨c, hc⟩ := nonempty_Ioo.mpr hx.1
    obtain ⟨d, hd⟩ := nonempty_Ioo.mpr hx.2
    have cdsub : [[c, d]] ⊆ Ioo (min a b) (max a b) := by
      rw [uIcc_of_le (hc.2.trans hd.1).le]
      exact Icc_subset_Ioo hc.1 hd.2
    replace hg_cont := hg_cont.mono (image_subset f cdsub)
    let J := [[sInf (f '' [[c, d]]), sSup (f '' [[c, d]])]]
    have hJ : f '' [[c, d]] = J := (hf.mono (cdsub.trans Ioo_subset_Icc_self)).image_uIcc
    rw [hJ] at hg_cont
    have h2x : f x ∈ J := by rw [← hJ]; exact mem_image_of_mem _ (mem_uIcc_of_le hc.2.le hd.1.le)
    have h2g : IntervalIntegrable g volume (f a) (f x) := by
      refine' hg1.mono_set _
      rw [← hf.image_uIcc]
      exact hf.surjOn_uIcc left_mem_uIcc (Ioo_subset_Icc_self hx)
    have h3g : StronglyMeasurableAtFilter g (𝓝[J] f x) :=
      hg_cont.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc (f x)
    haveI : Fact (f x ∈ J) := ⟨h2x⟩
    have : HasDerivWithinAt (fun u => ∫ x in f a..u, g x) (g (f x)) J (f x) :=
      intervalIntegral.integral_hasDerivWithinAt_right h2g h3g (hg_cont (f x) h2x)
    refine' (this.scomp x ((hff' x hx).Ioo_of_Ioi hd.1) _).Ioi_of_Ioo hd.1
    rw [← hJ]
    refine' (mapsTo_image _ _).mono _ Subset.rfl
    exact Ioo_subset_Icc_self.trans ((Icc_subset_Icc_left hc.2.le).trans Icc_subset_uIcc)