Adding app backgrounded result to rule compilation (#3533)

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

**Description**:

<!--
If at any point it isn't actively being worked on/ready for
review/otherwise moving forward strongly consider closing it (or not
opening it in the first place). If you decide not to close it, use Draft
PR while work is still in progress or use `DO NOT MERGE` label to
clarify the PRs state and comment with more information.
-->

**Steps to test this PR**:
1.
2.

<!--
Before submitting a PR, please ensure you have tested the combinations
you expect the reviewer to test, then delete configurations you *know*
do not need explicit testing.

Using a simulator where a physical device is unavailable is acceptable.
-->

**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)

Author: studiosutara

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