Is it normal to have chest pain and soreness after asthma attacks and coughing? My cough is productive and persistent. I am fatigued without fever.

Yes. If the cough has been severe and frequent, you may end up with chest wall pain. It appears that you are not over your asthma attack yet and you should talk to your doctor regarding adjusting your asthma treatment. Rarely rupture of air sacs (lung collapse) may cause similar symptoms.
May or may not be. After an exacerbation of asthma you may have some soreness of the chest. But if you continue to have symptoms such as cough, soreness we may have to suspect underlying pneumonia etc.
It's possible. It's not uncommon to have chest pain or discomfort associated with asthma or coughing. If your ribs/chest hurt when you push on it, it's a sign of muscle strain and may be helped with Ibuprofen. A productive cough can be associated with bronchitis or pneumonia as well. You should see a doctor to be examined to figure out more specifically what's going on & how best to treat you.

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