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When someone has sudden pain in the chest following a snap that occurs while coughing, it is most often due to a pull or strain of the muscles between the ribs. The primary risk for this is the intensity of the cough, not the underlying condition that is causing the cough, so can occur with any condition that can cause a stronger cough. Since you have been coughing phlegm for a week, it is most likely due to a bronchitis. In a non-smoker the most common cause of bronchitis is a viral infection, while smoking for years will increase the likelihood of a bacterial infection.
In this situation, there are a couple of approaches. It would be fine to take medicines to ease the musculoskeletal symptoms, and the primary interventions that can be done at home would be an over the counter anti-inflammatory medicine, such as ibuprofen or naproxen, as well as applying moist heat to the chest. It also can help to use medicines to help ease the cough, both the relieve ongoing physical stress on the muscles as well as limiting the risk for further injury. The best over the counter medicine would be a medicine to thin the phlegm, guaifenesin, combined with a cough suppressant, dextrmethorphan. This is available in many different products that are available over the counter, including generics.
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