Hard coughing can strain the tissues of the chest wall: ribs, cartilage or intercostal muscles. If severe, one can even sustain rib fractures.
The answer to the question depends on:
1) whether you feel healthy except for the cough; and
2) the intensity of the pain.
If cough is your only symptom, without fever or discolored sputum, you should try to suppress it with an over-the-counter cough medication. If you have fever or colored sputum or otherwise feel sick, you should see a doctor. If the pain below your ribs is mild and doesn't interfere with your sleep, more than 20 minutes, an over-the-counter analgesic should provide relief. On the other hand, if the pain is severe or gradually worsening, you should see your doctor and you will probably need an x-ray.
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