Adding Search Block to Bootstrap 3 Navbar
I had the same feature request.
In mytheme/template.php I've added

/*
 *  Preprocess page.tpl.php to inject the $search_box variable.
 */
function mytheme_preprocess_page(&amp;$variables){
  $search_form = drupal_get_form('search_form');
  $search_box = drupal_render($search_form);
  $variables['search_box'] = $search_box;
}

/*
 *  Form alter to add missing bootstrap classes and role to search form.
 */
function mytheme_form_alter(&amp;$form, &amp;$form_state, $form_id) {
  if ($form_id == 'search_form') {
    $form['#attributes']['class'][] = 'navbar-form';
    $form['#attributes']['role'][] = 'search';
  }
}


In mytheme/templates/page.tpl.php I've added

&lt;div class="col-md-3 nav nabvar-nav navbar-right"&gt;
  &lt;?php print $search_box; ?&gt;
&lt;/div&gt;


After the 

&lt;div class="navbar-header"&gt; div.


There still was a problem with a float:left applied to secondary links menu (instead of the proper boostrap navbar-right.
So, for a quick workaround, in mytheme/css/style.css I've added:

@media(max-width:768px){
  ul.secondary{
    float:none!important;
  }
}