If you connected your email inbox to HubSpot using IMAP, it's possible that the emails you send from contact, company, or deal records in HubSpot will not appear in your email client sent folder. When you send an email from HubSpot through your connected inbox, it's sent through your email's servers. In order for the email to appear in your sent folder, your email provider has to "copy" this email from the SMTP email server to your sent mail folder. Not all email providers do this automatically, so HubSpot can't guarantee that your email will appear in your email client sent folder.
If you're concerned about an issue with an email being sent successfully using your connected inbox, you can check to see if there is an error message on the contact record – emails sent successfully will display a status of sent on the email activity in the contact timeline.