Update Task to Closed Complete via REST API and 'Closed' and 'Closed By' Fields Not Updating

EDIT TO ORIGINAL POST: I have a strange issue where I am updating a task's state via the REST API to closed complete and it is successfully updating the task. I verify that it added a work note and the state is reflecting closed complete. However, when I search under closed tasks, the task is not present. I've also verified that the RITM and Request both reflect closed as well.

HERE IS THE KEY: I noticed that the 'Closed' and 'Closed_By' fields are not being automatically updated and are left blank. Which is why I am unable to locate the tasks in any closed task reports. When I first attempted to update some tasks I was only defining the 'Work_Notes' and 'State' fields be updated from the REST API. E.g.

# Specify Request Body
$Body1 = @{
'work_notes' = 'Closed via REST API'
'state' = '3'
}

But after I figured out the tasks were not reflecting closed in all of my reports due to the 'Closed' and 'Closed_by' fields being left blank, I tried to send those values down from REST API as well. E.g.

# Specify Request Body
$Body1 = @{
'work_notes' = 'Closed via REST API'
'closed_at' = '04-07-2018 03:19:46 PM'
'closed_by' = '6a207866130e16000080b027d144b0c6'
'state' = '3'
}

But that did not work and the 'Closed' and 'Closed By' fields still remain blank and thus are not reporting correctly. Is there a Business Rule that when you close a task in the UI it automatically applies the current authenticated user to 'Closed By' and the current date-time to 'Closed'? Perhaps that business rule does not apply when updating a task's state from the REST API?

Here is the entire Powershell script that I am using to update tasks through the ServiceNow REST API.

### Setup ServiceNow Auth and Headers
$SNowUser = "myusername"
$SNowPass = Cat "C:\ServiceNow\SecureString\Pass.txt" | ConvertTo-Securestring
$SNowCreds = New-Object —TypeName System.Management.Automation.PSCredential —ArgumentList $SNowUser, $SNowPass
$Headers = New-Object "System.Collections.Generic.Dictionary[[String],[String]]"
$Headers.Add('Accept','application/json')
$Headers.Add('Content-Type','application/json')

### Update SN Task State and Add Work-Notes
$SysID = "885f6824dbdd0380c1b87b6b8c961900"
# ServiceNow URI
$Uri1 = "https://myinstance.service-now.com/api/now/table/sc_task/$SysID"

# Specify HTTP Method
$Method1 = "PUT"

# Specify Request Body
$Body1 = @{
'work_note' = 'Test'
'state' = '3'
}

# Convert Body to JSON
$BodyJson1 = $Body1 | ConvertTo-Json

# Send HTTP Request ###
$Response1 = Invoke-WebRequest -Credential $SNowCreds -Headers $Headers -Method $Method1 -Uri $Uri1 -Body $BodyJson1 -ContentType "application/json" -UseBasicParsing
$JsonPUT1 = $Response1.Content | ConvertFrom-Json
$PUTObject1 = $JsonPUT1.result