Air taxi Captain reported the flight crew unknowingly entered a closed taxiway at LAX while trying to expeditiously exit the runway; and noted the closed taxiway was not outlined as closed on the airport charts; still had its signage up; and lacked any closure markings.

Narrative

I landed our aircraft; Aircraft X; on Runway 24R at Los Angeles International Airport (LAX). Our parking was located on the other side of the airport off of Runway 25L. We requested clearance for 25L from Approach Control but were denied due to operational constraints; requiring us to adapt to a longer taxi route.My First Officer (FO) and I briefed our taxi plan prior to landing; intending to extend our stopping distance to align with the published South Route from LAX's standard taxi routes. We reviewed the airport chart during our pre-landing briefing. No taxiway closures were outlined on the airport diagram regarding Runway 24R.During rollout; we passed Taxiway Uniform. At that moment; the Tower Controller issued an instruction to expedite our exit from the runway via Taxiway Victor. Having just passed Uniform; I identified the next taxiway ahead; which; according to the published airport chart; should have been Victor. A painted lead-off line from the runway guided us to this taxiway. I exited promptly to comply with the expedite instruction from the controller. Upon exiting; we were informed that the taxiway we took; Y; was closed and no longer in use. The controller directed us to re-enter the runway and exit via Victor. The closed Taxiway Y still had its sign up; though it was poorly lit and not clearly visible at night. Notably; the painted lead-off line to Taxiway Y lacked any closure markings; such as chevrons or red stop bars. The airport diagram and NOTAMs also did not notate any closures to taxiways for Runway 24R. For reference; Taxiway Y is located between U and V but is no longer on the chart.The following day I spoke with airport personnel. He acknowledged that many pilots have had issues exiting at the closed Taxiway Y; particularly at night; and that it is a known issue. He acknowledged that the Taxiway Y signs need to be removed to prevent future challenges.