Roots theme password protected post form function. Use the Bootstrap free markup with the Bootstrap LESS or use the Bootstrap markup version as standalone.

passwordform-bootstrap.php

<?php
  global $post;
  $label = 'pwbox-'.(empty($post->ID) ? rand() : $post->ID);
?>
<form action="<?php echo esc_url(site_url('wp-login.php?action=postpass', 'login_post')); ?>" class="post-password-form form-inline" method="post">
  <p><?php _e('This content is password protected. To view it please enter your password below:', 'roots'); ?></p>
  <p><label for="<?php echo $label; ?>" class="sr-only"><?php _e('Password:', 'roots'); ?></label></p>
  <div class="input-group">
    <input name="post_password" id="<?php echo $label; ?>" type="password" placeholder="<?php _e('Enter password', 'roots'); ?>" class="form-control" />
    <span class="input-group-btn">
      <button type="submit" class="post-password-submit btn btn-default"><?php _e('Submit', 'roots'); ?></button>
    </span>
  </div>
</form>

passwordform.php

<?php
  global $post;
  $label = 'pwbox-'.(empty($post->ID) ? rand() : $post->ID);
?>
<form action="<?php echo esc_url(site_url('wp-login.php?action=postpass', 'login_post')); ?>" class="post-password-form" method="post">
  <p><?php _e('This content is password protected. To view it please enter your password below:', 'roots'); ?></p>
  <p><label for="<?php echo $label; ?>"><?php _e('Password:', 'roots'); ?></label></p>
  <div>
    <input name="post_password" id="<?php echo $label; ?>" type="password" placeholder="<?php _e('Enter password', 'roots'); ?>" />
    <span>
      <button type="submit" class="post-password-submit"><?php _e('Submit', 'roots'); ?></button>
    </span>
  </div>
</form>

roots_get_the_password_form.php

function roots_get_the_password_form($output) {
  $output = '';
  locate_template('/templates/passwordform.php', true, false);
  return $output;
}
add_filter('the_password_form', 'roots_get_the_password_form');

roots_post_password_form.less

article {
	.post-password-form {
		> p > label {
			&:extend(.sr-only);
		}
		> div {
			&:extend(.input-group all);
			@media (min-width: @screen-sm-min) {
				display: inline-table;
				vertical-align: middle;
			}
			> span {
				&:extend(.input-group-btn all);
				@media (min-width: @screen-sm-min) {
					width: auto;
				}
			}
		}


		input[type=password] {
			&:extend(.form-control all, .input-group .form-control all);
			@media (min-width: @screen-sm-min) {
				width: 100%;
			}
		}
		.post-password-submit {
			&:extend(.btn all, .btn-default all, .input-group-btn:last-child > .btn, .input-group-btn > .btn all);
		}
	}
}