Expanded answer, was my first answer so excuse if there wasn’t enough detail before.
For Bootstrap 3.x I personally prefer CSS animations and I've been using animate.css & along with the Bootstrap Dropdown Javascript Hooks. Although it might not have the exactly effect you're after it's a pretty flexible approach.
Step 1: Add animate.css to your page with the head tags:
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/animate.css/3.4.0/animate.min.css">
Step 2: Use the standard Bootstrap HTML on the trigger:
<div class="dropdown">
  <button type="button" data-toggle="dropdown">Dropdown trigger</button>
  <ul class="dropdown-menu">
    ...
  </ul>
</div>
Step 3: Then add 2 custom data attributes to the dropdrop-menu element; data-dropdown-in for the in animation and data-dropdown-out for the out animation. These can be any animate.css effects like fadeIn or fadeOut
<ul class="dropdown-menu" data-dropdown-in="fadeIn" data-dropdown-out="fadeOut">
......
</ul>
Step 4: Next add the following Javascript to read the data-dropdown-in/out data attributes and react to the Bootstrap Javascript API hooks/events (http://getbootstrap.com/javascript/#dropdowns-events):
var dropdownSelectors = $('.dropdown, .dropup');
// Custom function to read dropdown data
// =========================
function dropdownEffectData(target) {
  // @todo - page level global?
  var effectInDefault = null,
      effectOutDefault = null;
  var dropdown = $(target),
      dropdownMenu = $('.dropdown-menu', target);
  var parentUl = dropdown.parents('ul.nav'); 
  // If parent is ul.nav allow global effect settings
  if (parentUl.size() > 0) {
    effectInDefault = parentUl.data('dropdown-in') || null;
    effectOutDefault = parentUl.data('dropdown-out') || null;
  }
  return {
    target:       target,
    dropdown:     dropdown,
    dropdownMenu: dropdownMenu,
    effectIn:     dropdownMenu.data('dropdown-in') || effectInDefault,
    effectOut:    dropdownMenu.data('dropdown-out') || effectOutDefault,  
  };
}
// Custom function to start effect (in or out)
// =========================
function dropdownEffectStart(data, effectToStart) {
  if (effectToStart) {
    data.dropdown.addClass('dropdown-animating');
    data.dropdownMenu.addClass('animated');
    data.dropdownMenu.addClass(effectToStart);    
  }
}
// Custom function to read when animation is over
// =========================
function dropdownEffectEnd(data, callbackFunc) {
  var animationEnd = 'webkitAnimationEnd mozAnimationEnd MSAnimationEnd oanimationend animationend';
  data.dropdown.one(animationEnd, function() {
    data.dropdown.removeClass('dropdown-animating');
    data.dropdownMenu.removeClass('animated');
    data.dropdownMenu.removeClass(data.effectIn);
    data.dropdownMenu.removeClass(data.effectOut);
    // Custom callback option, used to remove open class in out effect
    if(typeof callbackFunc == 'function'){
      callbackFunc();
    }
  });
}
// Bootstrap API hooks
// =========================
dropdownSelectors.on({
  "show.bs.dropdown": function () {
    // On show, start in effect
    var dropdown = dropdownEffectData(this);
    dropdownEffectStart(dropdown, dropdown.effectIn);
  },
  "shown.bs.dropdown": function () {
    // On shown, remove in effect once complete
    var dropdown = dropdownEffectData(this);
    if (dropdown.effectIn && dropdown.effectOut) {
      dropdownEffectEnd(dropdown, function() {}); 
    }
  },  
  "hide.bs.dropdown":  function(e) {
    // On hide, start out effect
    var dropdown = dropdownEffectData(this);
    if (dropdown.effectOut) {
      e.preventDefault();   
      dropdownEffectStart(dropdown, dropdown.effectOut);   
      dropdownEffectEnd(dropdown, function() {
        dropdown.dropdown.removeClass('open');
      }); 
    }    
  }, 
});
Step 5 (optional): If you want to speed up or alter the animation you can do so with CSS like the following:
.dropdown-menu.animated {
  /* Speed up animations */
  -webkit-animation-duration: 0.55s;
  animation-duration: 0.55s;
  -webkit-animation-timing-function: ease;
  animation-timing-function: ease;
}
Wrote an article with more detail and a download if anyones interested: 
article: http://bootbites.com/tutorials/bootstrap-dropdown-effects-animatecss
Hope that’s helpful & this second write up has the level of detail that’s needed
Tom