I laugh at him -- Indirect Object, Direct Object, or what?

Here is an interesting ESL question: What is "at him" in I laughed at him? I wasn't sure, so I asked the following question:

http://forum.wordreference.com/showthread.php?t=2447204&p=12309998

According to Wikipedia's Grammatical Object page,

There must be a direct object for an indirect object to be placed in a sentence.

In "They sent him a postcard", him is a (non-prepositional) indirect object of the verb sent (which uses a double-object construction). It typically corresponds to the dative case.
However, Wikipedia's transitivity page contradicts this with:

Even though an intransitive verb may not take a direct object, it often may take an appropriate indirect object: I laughed at him.

Which is accurate?

After much discussion, I wrote:

According to the book Ultimate Phrasal Verbs, there exist
"Transitive nonseparable phrasal verbs (verbs that require an object)... Hank's been CHEATING on his wife for years."

so maybe laugh is intransitive but laugh at is transitive.