Use cloudinary to convert animated gif to video and reduce file size. Learn more at http://cloudinary.com/blog/reduce_size_of_animated_gifs_automatically_convert_to_webm_and_mp4

convert-gif-to-video.php

<?php
add_filter( 'the_content', function ( $content ) {
	$doc = new DOMDocument();

	// modify state
	$libxml_previous_state = libxml_use_internal_errors( true );
	$doc->loadHTML( $content );
	// handle errors
	libxml_clear_errors();
	// restore
	libxml_use_internal_errors( $libxml_previous_state );

	$doc->preserveWhiteSpace = false;
	$images = $doc->getElementsByTagName( 'img' );
	$cloudinary_imgs = array();
	foreach ( $images as $img ) {
		$url = $img->getAttribute( 'src' );

		//if cloudinary image url
		if ( false !== stripos( $url, 'cloudinary' ) ) {
			$cloudinary_imgs[] = $img;
		}
	}

	$remove_images = array();
	foreach ( $cloudinary_imgs as $cloudinary_image ) {
		$url = $cloudinary_image->getAttribute( 'src' );

		if ( '.gif' === strrchr( $url, '.' ) ) {
			// store image object
			$remove_images[] = $cloudinary_image;

			// create video tag
			$video = $doc->createElement( 'video' );
			// add video tag attributes
			$video->setAttribute( 'autoplay', 'autoplay' );
			$video->setAttribute( 'loop', 'loop' );
			$video->setAttribute( 'muted', 'muted' );
			$video->setAttribute( 'poster',  $cloudinary_image->setAttribute( 'src', str_replace( '.gif', '.jpg', $url ) ); );

			// create source tag for mp4
			$mp4 = $doc->createElement( 'source' );
			$mp4->setAttribute( 'type', 'video/mp4' );
			$mp4->setAttribute( 'src', str_replace( '.gif','.mp4', $url ) );
			$video->appendChild( $mp4 );

			// create source tag for webm
			$webm = $doc->createElement( 'source' );
			$webm->setAttribute( 'type', 'video/webm' );
			$webm->setAttribute( 'src', str_replace( '.gif','.webm', $url ) );
			$video->appendChild( $webm );

			// create text message for browsers that doesn't support HTML5 video tag
			$no_video_support = $doc->createTextNode( 'Your browser does not support HTML5 video tag.' );
			$video->appendChild( $no_video_support );

			// create link to original gif animations for browsers that doesn't support HTML5 video tag
			$a = $doc->createElement( 'a' );
			$a->setAttribute( 'src', $url );
			$link_text = $doc->createTextNode( 'Click here to view original GIF' );
			$a->appendChild( $link_text );
			$video->appendChild( $a );

			// Add video tag to DOM
			$cloudinary_image->parentNode->insertBefore( $video, $cloudinary_image );
		}
	}
	// remove old .gif image tags
	foreach ( $remove_images as $rmimg ) {
		$rmimg->parentNode->removeChild( $rmimg );
	}

	return $doc->saveHTML();
}, 10, 1 );

Prerequisite, you've added the images using the Cloudinary WordPress plugin