system
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human
[Star Wars] Did Vader ever investigate or mention to Palpatine about the disappearance of Obi-wan's body when he tried to kill him

It seems odd for him to just brush it off when he's probably never seen anything like it happen from all the Jedi he's slain.
gpt
Given the relationship of Sith master and apprentice, it sets up a dynamic where Vader would be incentivized to hold this information secret.  Simply report that he slew Obi Wan on the Death Star.   

The moment would also have special significance to Vader.  There he was facing his old master, speaking about how the roles have changed.  At that moment, he also had to be thinking of how this was a step toward facing down his current master, the Emperor.  The combat against Obi Wan is like nothing he probably expected - a contest of wills more then lightsabers, ending in Obi Wan's seeming acceptance of his death with some vague threat about becoming more powerful?  What?  This might have thrown Vader in a tailspin about his eventual conflict with the Emperor.

Its interesting that when that conflict did come, it was Vader's turn to do the unexpected.  To also accept his death, but become more powerful (in a way) than the Emperor could have expected. 

Also, what did happen to Obi Wan?  Neither being a visible specter nor whispering advice are in any way more powerful than a living Obi Wan was to Luke? A living Obi Wan could have been immensely influential in shaping Luke, and therefore the course of history?  What did happen to him?

I suspect he was able to join, at least for a time, with the force and shape events to aid Luke in ways a living Obi Wan could not.  The destruction of Alderaan probably impressed on him the stakes. It was Obi Wan who probably helped that torpedo turn 90 degrees down a port to blow up the Death Star.  Maybe Obi Wan influenced Han Solo to come back to help at that pivotal moment.  Maybe Obi Wan even helped Darth Vader to become possibly the sole survivor of the Death Star, as odd as that sounds.   Obi Wan, dead, may have been the big hero of Yavin, acting mostly through Luke.