This randomized phase II trial studies how well ruxolitinib phosphate, and bosutnib, dasatinib, imatinib or nilotinib, work in treating patients with chronic myeloid leukemia. Chronic myeloid leukemia cells produce a protein called BCR-ABL. The BCR-ABL protein helps chronic myeloid leukemia cells to grow and divide. Tyrosine kinase inhibitors, such as bosutinib, dasatinib, and nilotinib, stop the BCR-ABL protein from working, which helps to reduce the amount of chronic myeloid leukemia cells in the body. Ruxolitinib is a different type of drug that helps to stop the body from making substances called growth factors. Chronic myeloid leukemia cells need growth factors to grow and divide. The addition of ruxolitinib to the tyrosine kinase inhibitor may or may not help reduce the amount of chronic myeloid leukemia cells in the body.

PRIMARY OBJECTIVE:

I. To compare the rate of molecular response 4.5 (MR4.5) after 12 months of combination therapy with ruxolitinib phosphate (ruxolitinib) plus a tyrosine-kinase inhibitor (TKI) (bosutinib, dasatinib, imatinib or nilotinib) versus a TKI alone, based on local polymerase chain reaction (PCR) testing to measure BCR-ABL transcripts in chronic phase chronic myelogenous leukemia (CML) patients with molecular evidence of disease.

SECONDARY OBJECTIVES:

I. To estimate the frequency and severity of toxicities of each regimen in this patient population.

II. To estimate progression free survival and overall survival of each regimen in this patient population.

ADDITIONAL OBJECTIVES:

I. To describe patterns of MR4.5 and molecular response 4.0 (MR4.0) attainment and failure over the 3, 6, 9, and 12-month time points of each regimen in this patient population.

II. To evaluate drug compliance based on patient reported drug intake calendars in this patient population.

III. To describe the kinetics of response in this patient population (as measured by quantitative BCR-ABL/BCR ratio) in both arms over the 3, 6, 9, and 12-month time points.

OUTLINE: Patients are randomized to 1 of 2 arms.

ARM I: Patients receive bosutinib orally (PO) daily or dasatinib PO daily or nilotinib PO twice daily (BID) or imatinib PO daily on days 1-90. Treatment repeats every 90 days for up to 4 cycles in the absence of disease progression or unacceptable toxicity.

ARM II: Patients receive ruxolitinib phosphate PO BID on days 1-90, and bosutinib PO daily or dasatinib PO daily or nilotinib PO BID or imatinib PO daily on days 1-90. Treatment repeats every 90 days for up to 4 cycles in the absence of disease progression or unacceptable toxicity.

After completion of study treatment, patients are followed up every 6 months for 2 years, and then annually up to 5 years.

Inclusion Criteria:

• Patients must have a diagnosis of chronic phase chronic myeloid leukemia without any history of progression to accelerated or blast phase CML; no new bone marrow aspiration and biopsy is needed to prove diagnosis prior to randomization; however, documentation stating the patient is in chronic phase is required
• Patients must have detectable BCR-ABL transcripts measured by reverse transcriptase (RT)-PCR at a clinical laboratory improvement act (CLIA)-approved laboratory and reported on the international scale (IS) with a value of > 0.0032% IS and =< 1.0% IS within 21 days prior to randomization; the RT-PCR assay must have the sensitivity to detect a 4.5 log reduction in BCR-ABL transcripts from 100% IS (0.0032% IS or lower)
• Patients must have been receiving TKI treatment for CML for at least 12 months prior to randomization. Hydroxyurea prior to initiation of TKI is allowed.
• Patients must be currently receiving treatment with bosutinib (within the allowable dose range of 200-500 mg daily), nilotinib (within the allowable dose range of 150-400 mg BID or a cumulative daily dose of 300-800 mg), dasatinib (within the allowable dose range of 40-140 mg daily), or imatinib (within the allowable dose range of 300-400 mg daily); they must have received their current TKI for a minimum of 6 months prior to randomization and must be expected to remain on the same TKI for the next 12 months
• Patient must not have a history of resistance to any prior TKI drug; if patient has received more than one TKI, the reason for changing treatment must have been something other than resistance or inadequate response to the prior TKI (for example, intolerance to the prior TKI) and the treatment change must have occurred >= 6 months prior to randomization.
• Patients must not be receiving any other investigational agents
• Patients must have complete history and physical examination within 28 days prior to randomization
• If clinically indicated, patients must have corrected Fridericia's correction formula (QTcF) interval < 500 ms (by Fridericia calculation) on a 12-lead electrocardiography (EKG) within 7 days prior to randomization
• Platelets >= 100,000/mm^3 (100.0 x 10^9/L) within 7 days prior to randomization
• Absolute neutrophil count (ANC) > 1,000/mm^3 (1.0 x 10^9/L) within 7 days prior to randomization
• Hemoglobin >= 8 g/dL within 7 days prior to randomization
• Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =< 2.5 x institutional upper limit of normal (IULN) within 7 days prior to randomization
• Total bilirubin =< 1.5 x IULN within 7 days prior to randomization (unless the patient has a known diagnosis of Gilbert's syndrome)
• Serum creatinine =< 1.5 x IULN within 7 days prior to randomization
• Prior malignancy is allowed providing it does not require concurrent therapy; exception: active hormonal therapy is allowed
• Patients must not be pregnant or nursing due to the teratogenic potential of the drugs used on this study; women of child-bearing potential must have a negative serum pregnancy test within 7 days prior to randomization; women/men of reproductive potential must have agreed to use an effective contraceptive method during treatment and for 30 days after discontinuation of study drug; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures
• Patients known to be human immunodeficiency virus positive (HIV+) are eligible provided they meet all other eligibility criteria and have undetectable HIV viral loads on their most recent viral load test which must have been performed in the last 6 months
• Specimens (peripheral blood) must be collected and submitted to a CLIA-approved laboratory, within 21 days prior to randomization; BCR-ABL transcripts must be measured using RT-PCR and results must be reported using the international scale; the RT-PCR assay must have the sensitivity to detect a 4.5 log reduction in BCR-ABL transcripts from 100% IS (must be able to detect 0.0032% IS or lower)
• Patients must be offered participation in submission of specimens for central BCR-ABL quantification and banking for future specimens; this submission is highly encouraged as an important protocol endpoint; with patient's consent, specimens must be collected and submitted, within 21 days prior to randomization
• Patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines
• As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system