Inclusion Criteria:

          -  Participants with histologically or cytologically documented diffuse large B-cell
             lymphoma (DLBCL) must meet at least 1 of the following risk criteria:

               -  Age-adjusted International Prognostic Index (IPI) score: 2-3

               -  Ki-67 >= 80%

               -  Histologically, or cytologically documented activated B-cell-like (ABC, also
                  known as post-GCB) subtype

               -  Double hit variant, defined as having v-myc avian myelocytomatosis viral oncogene
                  homolog (MYC) gene rearrangement in the presence of B-cell chronic lymphocytic
                  leukemia (CLL)/lymphoma (BCL) 2 or BCL6 gene rearrangement

                    -  Other aggressive non-DLBCL non-Burkitt non-Hodgkin B-cell lymphoma variants
                       as defined by the 2008 World Health Organization (WHO) classification,
                       including rare CD20 negative B-cell lymphomas (i.e. plasmablastic lymphoma,
                       and primary effusion lymphoma) are also eligible; grade 3B follicular
                       lymphoma is also eligible as long as one the above risk criteria is met

          -  Participants who are untreated or who received a maximum of one (1) cycle of
             combination chemotherapy, including rituximab-containing regimens, prior are eligible;
             the start of previous chemotherapy cycle must occur at least 21 days prior to
             beginning treatment under this protocol, and such cycle will count towards the total
             maximum of 6 cycles under this study

          -  Documentation of HIV infection at any time prior to study entry; documentation may be
             molecular (detectable viral ribonucleic acid [RNA] by polymerase chain reaction
             [PCR]), serologic (positive enzyme-linked immunosorbent assay [ELISA] and positive
             Western blot), or other federally approved licensed HIV test; prior documentation of
             HIV seropositivity is acceptable

          -  All stages of disease

          -  Measurable or non-measurable tumor parameter(s); non-measurable tumor parameters are
             defined as not having bidimensional measurements (e.g., gastric or marrow
             involvement), but can be followed for response by other diagnostic tests such as
             gallium, positron emission tomography (PET) imaging, and/or bone marrow biopsy

          -  Performance status (PS) 0, 1, or 2 per the Eastern Cooperative Oncology Group (ECOG)
             performance status scale (Karnofsky performance score >= 50%)

          -  Able to provide informed consent

          -  Total bilirubin =< 1.5 institutional upper limit of normal (ULN), unless elevated
             secondary to lymphomatous involvement of liver or biliary system, or due to other HIV
             medications (e.g., indinavir, tenofovir, or atazanavir); for direct bilirubin > 1.2
             due to hepatic involvement by tumor for the initial dose of EPOCH drug adjustment

          -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =< 2.5 times ULN
             (unless elevated due to secondary lymphomatous involvement of the liver)

          -  Negative for hepatitis B, or if infected with hepatitis B, receiving anti-hepatitis B
             therapy; all participants will be required to be screened for hepatitis B and C; per
             Infectious Disease Society of America (IDSA) and American Association for the Study of
             Liver Diseases (AASD) guidelines, those participants that show no immunity, defined by
             the lack of hepatitis B surface antibody, and show evidence of chronic infection (i.e.
             hepatitis B surface antigen positive [HBsAg+], hepatitis B core antibody positive
             [HBcore+], hepatitis surface antibody negative [HBsAB-]) will be required to be on
             anti-hepatitis B therapy, during the study, in order to be eligible; participants will
             be permitted to enroll in the study provided liver function tests meet criteria, and
             there is no evidence of cirrhosis; the exact hepatitis B therapy will be at the
             discretion of the infection disease specialist or investigator; however all
             participants who present with acute hepatitis B or show normal transaminases and are
             HBsAg+ and immunoglobulin (Ig)M+ for hepatitis core antigen will not be eligible for
             trial enrollment; participants who are hepatitis C antibody positive, with or without
             a positive hepatitis C RNA level, will be permitted to enroll in the study provided
             liver function tests meet criteria, and have no evidence of cirrhosis; participants
             diagnosed with hepatitis C less than 6 months from trial enrollment, will be
             considered to have acute hepatitis C and will be excluded from study unless hepatitis
             (hep) C viral load is undetectable

          -  Creatinine clearance >= 60 mL/min, unless secondary to renal involvement by lymphoma;
             for creatinine clearance < 50 mL/min due to kidney involvement by tumor

          -  Granulocytes/absolute neutrophil count (ANC) >= 1,000/mm^3

          -  Platelets >= 75,000/mm^3 (unless these parameters are abnormal secondary to
             lymphomatous involvement of bone marrow); all participants must cease
             colony-stimulating factor therapy at least 24 hours prior to institution of cycle 1
             chemotherapy

          -  Left ventricular ejection fraction (LVEF) that is at or above the lower institutional
             limits of normal, as assessed by multiple gated acquisition (MUGA) scan or
             echocardiogram within the 6 weeks prior to registration

          -  Concurrent radiation, with or without steroids, or steroids alone for emergency
             conditions secondary to lymphoma (i.e. cord compression, etc.) will be permitted

          -  Female participants must have a negative pregnancy test within 7 days of entering into
             the study; both men and women of child bearing potential must agree to use adequate
             methods of contraception for the duration of the treatment; women must avoid
             pregnancy, and men must avoid fathering children while in the study and for 6 months
             following the last study drug treatment

          -  Participants on an antiretroviral regimen should be receiving treatment that is in
             accordance with the current International acquired immune deficiency syndrome (AIDS)
             Society guidelines; the specific agents are at the discretion of the investigator and
             use of agents currently available on an expanded access basis is allowed but use of
             experimental antiretroviral agents or those containing zidovudine (including Combivir
             and Trizivir) are prohibited; changes to highly active anti-retroviral therapy (HAART)
             therapy may be made if medically necessary (toxicity, failure of regimen, etc.);
             antiretroviral naïve participants: participants who are not on HAART at study entry
             MUST begin therapy (utilizing the above guidelines) AFTER one cycle of chemotherapy
             has been completed under protocol; changes to HAART therapy may be made if medically
             necessary (toxicity, failure of regimen, etc.); concurrent therapy with zidovudine or
             a zidovudine-containing regimen (including Combivir and Trizivir) will be prohibited
             until 2 months following the participant's completion of chemotherapy as part of this
             protocol; the use of cobicistat (e.g., Tybost), or cobicistat containing single tablet
             regimens (e.g., Stribild) is prohibited during concurrent chemotherapy under this
             protocol; participants taking cobicistat or cobicistat-containing single table
             regimens must switch to a different agent or regimen prior to enrollment, and will
             remain on the regimen until at least 2 months following treatment discontinuation;
             Cobicistat is a pure and potent cytochrome P450, family 3, subfamily A, polypeptide 4
             (CYP3A4) inhibitor and has the potential to increase the area under the curve (AUC) of
             CYP3A4 substrates; therefore, both vincristine and doxorubicin would have the
             potential for drug drug interaction (DDI) with cobicistat since they are CYP3A4
             substrates

          -  Participants already receiving erythropoietin or colony-stimulating factor therapy are
             eligible for participation, although the latter must be discontinued at least 24 hours
             prior to receiving chemotherapy

          -  Participants must be able to swallow oral medications

        Exclusion Criteria:

          -  Participants who have received more than one (1) prior cycle of chemotherapy similar
             to cyclophosphamide, doxorubicin hydrochloride, vincristine sulfate, and prednisone
             (CHOP) or EPOCH with or without rituximab

          -  Absolute CD4 count of < 50 cells/ mm^3

          -  Presence of second active tumor, other than non-melanoma skin cancer, carcinoma in
             situ of the cervix, or Kaposi's sarcoma (KS) that requires systemic therapy

          -  Central nervous system (CNS) involvement by lymphoma including parenchymal brain or
             spinal cord lymphoma or known presence of leptomeningeal disease prior to registration

          -  Participants with viral hepatitis who do not meet the criteria will not be eligible;
             all participants who present with acute hepatitis B including those with normal
             transaminases who are HBsAg+ and IgM + for hepatitis core antigen will not be
             eligible; participants who are hepatitis B core antibody positive are eligible only if
             they start or are on prophylactic therapy; a hepatitis B viral load should be
             confirmed negative on all participants who are hepatitis B core antibody positive, but
             hepatitis B antigen negative; participants refusing to take any anti-hepatitis B
             therapy during study will also be excluded; participants diagnosed with hepatitis C
             are eligible if they meet criteria

          -  Pregnant women or nursing mothers

          -  ECOG performance score >= 3 (Karnofsky performance status [KPS] < 50%)

          -  Expected survival < 2 months

          -  Unable to comply with the requirements of the protocol, or unable to provide adequate
             informed consent in the opinion of the principal investigator

          -  Serious, ongoing, non-malignant disease or infection, which in the opinion of the
             investigator and/or the sponsor would compromise other protocol objectives;
             participants with active opportunistic infections are ineligible

          -  Major surgery, other than diagnostic surgery, occurring 4 weeks prior to study entry;
             splenectomy will not be considered an exclusionary major surgery

          -  Rituximab therapy within the 12 months prior to study entry; participants treated with
             rituximab within 12 months prior to study registration are eligible only if it was
             given for indications other than the treatment of aggressive lymphoma

          -  Prior cytotoxic chemotherapy or radiotherapy for this lymphoma

          -  History of cutaneous or mucocutaneous reactions, or diseases in the past, due to any
             cause, severe enough to cause hospitalization or an inability to eat or drink for > 2
             days; this exclusion relates to the long-term possibility of severe cutaneous or
             mucocutaneous reactions to rituximab that might occur at increased frequency in
             participants who have had severe skin disease or reactions in the past

          -  Use of zidovudine or cobicistat as part of the HAART regimen (a drug substitution at
             the time of study entry is allowed)

          -  Any acute, inter-current infection that may interfere with planned protocol treatment;
             participants with mycobacterium avium will not be excluded from study entry; chronic
             therapy with potentially myelosuppressive agents is allowed provided that entry
             hematologic criteria are met

          -  Myocardial infarction (MI) within 6 months prior to study entry, New York Heart
             Association (NYHA) class II or greater heart failure, uncontrolled angina, severe
             uncontrolled ventricular arrhythmias, clinically significant pericardial disease, or
             electrocardiographic evidence of acute ischemic or active conduction system
             abnormalities

          -  Participants should not have taken valproic acid or another histone deacetylase
             inhibitor for at least 2 weeks prior to study enrollment