Integrase and Maraviroc Intensification in Neurocognitive Dysfunction (InMIND)

Inclusion Criteria: * HIV-1 infection, documented by: * a licensed rapid HIV test or HIV enzyme or chemiluminescence immunoassay (E/CIA) test kit at any time prior to study entry and confirmed by a licensed Western blot or a second antibody test by a method other than the initial rapid HIV and/or E/CIA, or by HIV-1 antigen or plasma HIV-1 RNA viral load. NOTE: The term "licensed" refers to a United States Food and Drug Administration (FDA)-approved kit, which is required for all IND studies, or for sites located in countries other than the United States, a kit that has been certified or licensed by an oversight body within that country and validated internally. Non-US sites are encouraged to use US FDA-approved methods for IND studies. WHO (World Health Organization) and CDC (Centers for Disease Control and Prevention) guidelines mandate that confirmation of the initial test result must use a test that is different from the one used for the initial assessment. A reactive initial rapid test should be confirmed by either another type of rapid assay or an E/CIA that is based on a different antigen preparation and/or different test principle (eg, indirect versus competitive), or a Western blot or a plasma HIV-1 RNA. OR * Documentation of HIV diagnosis in the medical record by a healthcare provider. * On current ART for at least 6 months prior to study entry with no interruption in treatment of greater than or equal to 7 consecutive days. Note: The following ART changes are allowed: * Tenofovir disoproxil fumarate (TDF) to tenofovir alafenamide fumarate (TAF)/TAF-containing fixed-dose combination regimens * Ritonavir (RTV) to cobicistat (COBI)/COBI-containing fixed-dose combination regimens * No plans to change ART while on study. Note: The following planned ART changes are allowed: * TDF to TAF/TAF-containing fixed-dose combination regimens * RTV to COBI/COBI-containing fixed-dose combination regimens * HIV-1 plasma RNA less than 50 copies/mL obtained within 90 days prior to study entry by any FDA-approved assay at any United States laboratory that has a Clinical Laboratory Improvement Amendments (CLIA) certification or its equivalent, or at any network-approved non-US laboratory that operates in accordance with Good Clinical Laboratory Practices (GCLP) and participates in appropriate external quality assurance programs. * No more than one HIV-1 plasma RNA greater than or equal to 50 and less than 200 copies/mL (only one "blip") in the past 6 months with a subsequent HIV-1 plasma RNA less than 50 copies/mL. NOTE: There should be no plasma HIV-1 RNA greater than 200 copies/mL within the 6 months prior to study entry. * HAND diagnosis (ANI, MND, or HAD) within 60 days prior to study entry. HAND is defined as at least mild impairment on neurocognitive testing (more than one standard deviation below appropriate normative data in two domains of functioning) and no severely confounding factors. * Screening laboratory values obtained within 60 days prior to study entry by any US laboratory that has a CLIA certification or its equivalent, or at any network-approved non-US laboratory that operates in accordance with GCLP and participates in appropriate external quality assurance programs: * Absolute neutrophil count (ANC) greater than or equal to 500/mm\^3 * Hemoglobin greater than or equal to 7.5 g/dL * Platelet count greater than or equal to 40,000/mm\^3 * Creatinine less than or equal to 2.0 x upper limit of normal (ULN) * Aspartate transaminase (AST) less than or equal to 5 x ULN * Alanine transaminase (ALT) less than 3 x ULN * Alkaline phosphatase less than or equal to 5 x ULN * Total bilirubin less than 1.5 x ULN. NOTE: If the potential participant is taking an indinavir (IDV)- or atazanavir (ATV)-containing regimen at the time of screening, total bilirubin less than or equal to 5 x ULN is acceptable. * Creatinine clearance (CrCl) greater than or equal to 60 mL/min, either measured or estimated by Cockcroft-Gault equation. NOTE: A calculator for estimating the CrCl can be found at www.fstrf.org/ACTG/ccc.html * Females of reproductive potential (women who have not been post-menopausal for at least 24 consecutive months, ie, who have had menses within the preceding 24 months, or women who have not undergone surgical sterilization, hysterectomy or bilateral salpingectomy or bilateral oophorectomy or tubal ligation) must have a negative serum or urine pregnancy test by any US clinic or laboratory that has a CLIA certification or its equivalent, or is using a point of care (POC) / CLIA-waived test, or at any network-approved non-US laboratory or clinic that operates in accordance with GCLP and participates in appropriate external quality assurance programs within 48 hours prior to study entry * Females of reproductive potential must agree not to participate in the conception process (ie, active attempt to become pregnant, in vitro fertilization), and if participating in sexual activity that could lead to pregnancy, must use at least one reliable form of contraception. Female participants must use contraceptives while receiving study treatment and for 6 weeks after stopping study treatment. More information on this criterion is available in the protocol. * Ability and willingness of participant to complete the neuropsychological tests * Ability and willingness of participant or a legally authorized representative (see protocol for more information) to provide informed consent * Ability and willingness to take oral study medications Exclusion Criteria: * Current or past medical condition(s) that in the opinion of the investigator prevents attribution of the cause of cognitive impairment to HIV. For example: * Major depressive disorder with psychotic features * Traumatic Brain Injury (TBI) with a clear impact on activities of daily living * Developmental delay, intellectual deficit, and/or severe educational disability resulting in some dependence for activities of daily living * Ongoing substance use disorder with significant impact on activities of daily living. Difficult or impossible to determine whether cognitive or functional decline is due to substance use or HIV, or both * Evidence of intoxication or withdrawal during the screening evaluation * Central nervous system (CNS) infections or opportunistic conditions: brain abscess (bacterial, mycobacterial, fungal or Toxoplasma), meningitis with persistent neurologic impairment, primary CNS lymphoma, progressive multifocal leukoencephalopathy (PML), or another structural brain lesion with neurological sequelae * Other CNS conditions: non-opportunistic primary or metastatic brain tumors, uncontrolled seizure disorder, progressive multiple sclerosis, stroke with neurological sequelae, or dementia due to causes other than HIV (eg, Alzheimer's disease) * Constitutional illness (eg, persistent unexplained fever, diarrhea, significant weight loss, disabling weakness) within 30 days of screening * Known untreated B12 deficiency or malnutrition (body mass index \[BMI\] less than 18) at screening * Evidence of current hepatitis C virus infection (HCV) (ie, HCV antibody \[Ab\] positive within 90 days prior to study entry unless also shown to be plasma HCV RNA negative within the same time period) * Unstable and advanced liver disease (as defined by the presence of at least one of the following: ascites, encephalopathy, coagulopathy, hypoalbuminemia, esophageal or gastric varices, or persistent jaundice) * Prior or current use of any CCR5 antagonist (such as MVC and cenicriviroc \[CVC\]) and integrase inhibitor (such as RAL, DTG, and elvitegravir \[EVG\]) * Current use of any medication, including antiretrovirals, prohibited in the study (refer to the A5324 protocol-specific web page \[PSWP\] for the prohibited medications) * Breastfeeding * Presence of an AIDS-defining opportunistic infection within 6 months prior to entry. Note: Refer to the A5324 Manual of Operations (MOPS) for the list of AIDS-defining opportunistic infections. * Active syphilis or treatment for syphilis within 90 days prior to study entry. NOTE: Active syphilis is defined as four-fold increase in serum rapid plasma reagin (RPR) or venereal disease research laboratory (VDRL) tests in an individual with past syphilis, or newly reactive serum RPR or VDRL with a reactive confirmatory test (enzyme immunoassays \[EIA\] or chemiluminescent assay \[CIA\], T. pallidum particle agglutination \[TP-PA\], or fluorescent treponemal antibody absorbed \[FTA-ABS\]). * Known allergy/sensitivity or any hypersensitivity to components of study drugs or their formulation