Multi-page posts do not get indexed by Google due to canonical URLs

You will need to replace the default rel_canonical function to do this:

function wpse_84647_rel_canonical() {

    /* Only affect singular posts. Exit early if Yoast SEO is doing this for us */
    if ( ! is_singular() || class_exists( 'WPSEO_Frontend' ) )
        return;

    /* Get the post permalink */
    $post = get_queried_object();
    $link = trailingslashit( get_permalink( $post->ID ) );

    /* Get the number of pages the post is split into */
    $numpages = substr_count( $post->post_content, '<!--nextpage-->' ) + 1;

    /* Get the current pagination page number */
    $page = ( get_query_var( 'page' ) ? get_query_var( 'page' ) : 1 );

    /* Add the page number if the post is paginated */
    if ( $numpages && $page > 1 )
        $canonical = user_trailingslashit( $link . $page );
    else
        $canonical = get_permalink( $post->ID );

    /* Output the canonical link */
    printf ( "\n" . '<link rel="canonical" href="https://wordpress.stackexchange.com/questions/84647/%s" />', $canonical );

    /* Add the adjacent rel links */
    if ( $page > 1 ) {
        remove_action( 'wp_head', 'adjacent_posts_rel_link_wp_head', 10, 0 );
        /* Link to the previous adjacent post */
        $prev = $page - 1;
        if ( $prev > 1 )
            printf ( "\n" . '<link rel="prev" href="https://wordpress.stackexchange.com/questions/84647/%s" />', user_trailingslashit( $link . $prev ) );
        else
            printf ( "\n" . '<link rel="prev" href="https://wordpress.stackexchange.com/questions/84647/%s" />', get_permalink( $post->ID ) );
    }
    if ( $page < $numpages ) {
        remove_action( 'wp_head', 'adjacent_posts_rel_link_wp_head', 10, 0 );
        /* Link to the next adjacent post */
        $next = $page + 1;
        printf ( "\n" . '<link rel="next" href="https://wordpress.stackexchange.com/questions/84647/%s" />', user_trailingslashit( $link . $next ) );
    }
    print "\n";
}

remove_action( 'wp_head', 'rel_canonical' );
add_action( 'wp_head', 'wpse_84647_rel_canonical', 9 );

This new function adds gets the paginated page from the URL and adds it to the canonical link. It will also add next and previous links to the adjacent pages so Google knows

Also, if you use Yoast’s WordPress SEO plugin you will not need this snippet, as the plugin handles canonical links for you.