How to sync iOS contact birthdays with Google Calendar?

I'm trying to make my ios contact birthdays show up on my google calendar on my mac. They show up on the google calendar app on my iphone, but not on the website on my mac. I've already tried sharing through the icloud website, ie making my birthdays calendar public, but it doesn't work because apparently I'm subscribed to my own birthdays calendar rather than owning it. I know I can export my birthdays calendar from ios and import it into google calendar, but I'm trying to avoid that so I can have a birthdays calendar that updates when I put new birthdays into my contacts.

[Re-Titled by Moderator]