What is short code for following?

You are correct, [login_form] is the shortcode that you should use. From the Shortcode API

The add_shortcode function is used to register a shortcode handler. It takes two parameters: the shortcode name (the string used in a post body), and the callback function name.

From what I can understand is that the shortcode calls a customized login form defined in a function called pippin_login_form_fields(); There might be a couple of reasons that your shorcode does not display anything

  • You are logged in. The shortcode first does a check to see if a user is logged in or not (if(!is_user_logged_in()) {). If a user is not logged in, the login form is displayed. When a user is logged in, nothing is displayed.

  • There might be some type of error in the pippin_login_form_fields(); function or it is misplaced/missing

  • Some other function or filter is breaking your shortcode.

If point one is not the problem, turn on debugging in wp-config.php and check for any errors. To debug wordpress, go and check this article: Debugging in WordPress