Installing compact search in the main menu?

You can create a menu item for search icon/button and check it’s ID in HTML code. Then use can use jQuery click function to show and hide search div.

For example if my menu item list (li) id is menu-item-22 and search form container class is search_div then this will be the jQuery for displaying/hiding search form on click.

$('#menu-item-22 a').click(function(e) {
    e.preventDefault();
    $('.search_div').toggle();
});

Also, positioning of search form container will be defined in CSS.