Adding app backgrounded result to rule compilation (#3533)

Task/Issue URL:
https://app.asana.com/0/1204556816597738/1208691511506504/f
Tech Design URL:
CC:

**Description**:

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**Definition of Done (Internal Only)**:

* [ ] Does this PR satisfy our [Definition of
Done](https://app.asana.com/0/1202500774821704/1207634633537039/f)?

**Copy Testing**:

* [ ] Use of correct apostrophes in new copy, ie `’` rather than `'`

**Orientation Testing**:

* [ ] Portrait
* [ ] Landscape

**Device Testing**:

* [ ] iPhone SE (1st Gen)
* [ ] iPhone 8
* [ ] iPhone X
* [ ] iPhone 14 Pro
* [ ] iPad

**OS Testing**:

* [ ] iOS 15
* [ ] iOS 16
* [ ] iOS 17

**Theme Testing**:

* [ ] Light theme
* [ ] Dark theme

---
###### Internal references:
[Software Engineering
Expectations](https://app.asana.com/0/59792373528535/199064865822552)
[Technical Design
Template](https://app.asana.com/0/59792373528535/184709971311943)

Core/PixelEvent.swift

@@ -1729,6 +1729,7 @@ extension Pixel.Event {
 
         case tabClosed = "tab_closed"
         case appQuit = "app_quit"
+        case appBackgrounded = "app_backgrounded"
         case success
 
     }

DuckDuckGo/RulesCompilationMonitor.swift

@@ -41,6 +41,11 @@ final class RulesCompilationMonitor {
                                                selector: #selector(applicationWillTerminate(_:)),
                                                name: UIApplication.willTerminateNotification,
                                                object: nil)
+
+        NotificationCenter.default.addObserver(self,
+                                               selector: #selector(applicationDidEnterBackground),
+                                               name: UIApplication.didEnterBackgroundNotification,
+                                               object: nil)
     }
 
     /// Called when a Tab is going  to wait for Content Blocking Rules compilation
@@ -88,6 +93,18 @@ final class RulesCompilationMonitor {
 
         reportWaitTime(CACurrentMediaTime() - waitStart, result: .appQuit)
     }
+
+    /// If App is going into the background while the rules are still being compiled, report the time so that we
+    /// do not continue to count the time in background
+    @objc func applicationDidEnterBackground() {
+        guard !didReport,
+              !waiters.isEmpty,
+         let waitStart = waitStart
+        else { return }
+
+        reportWaitTime(CACurrentMediaTime() - waitStart, result: .appBackgrounded)
+    }
+
 
     private func reportWaitTime(_ waitTime: TimeInterval, result: Pixel.Event.CompileRulesResult) {
         didReport = true