Atrial fibrillation (AF) is the most common arrhythmia encountered in clinical practice. Previous studies have shown that rate control strategy in AF is non-inferior to rhythm control strategy, in terms of stroke and mortality risk. In addition, rate control strategy is associated with lower risk of hospitalization and non-cardiovascular mortality. Therefore, rate control is an essential strategy to improve quality of life, decrease morbidity and prevent tachycardia-induced cardiomyopathy in AF patients. The recommended rate control agents include beta-blocker, nondihydropyridine calcium-channel blocker (CCB), digoxin and amiodarone. However, in heart failure with reduced ejection fraction patient, the medication of rate control were beta-blocker than digoxin. But several clinical observation study show excess mortality in AF patients. Ivabradine, a If inhibitor, it is well-established that a pacemaker current, If current, is functionally expressed in the sinus node . Previous studies have shown that Ivabradine, If inhibitor, significantly reduces sinus rate and improves prognosis in patients with systolic heart failure. Interestingly, several investigators found that hyperpolarization- activated cyclic nucleotidylated channel 4 current (HCN4), the main isoform of the channel responsible for If current, is also functionally expressed in the Atrioventricular node(AV node). Recent data have shown that inhibition of If current slows AV node conduction in animals and humans. Thus, we want to compare the effect of Ivabradine on ventricular rate with digoxin in this study.

Inclusion Criteria:

1. The patient with persistent or permanent atrial fibrillation(24hr ECG) with HFrEF and HFmrEF.
2. After maximal tolerance dose beta-blocker(Bisoprolol 20mg/day，Carvedilol 50mg/day) or intolerant to beta-blocker the resting heart rate from ECG is still faster than 100 or resting heart rate from ECG is greater than 80 but still with symptoms of short of breath and palpitation.
3. Stable heart rhythm medication.(no change of medication in recently one week)
4. Age 20 to 90 years old.
5. The subject must be an adult who can read himself/herself and walk independently.

Exclusion Criteria:

1. Used medication with interaction with digoxin : Clarithromycin, Erythromycin, Azithromycin, ritonavir, lopinavir/ritonavir, Doxycycline, Minocycline, tetracycline。
2. Used medication with interaction with ivabradine: voriconazole, posaconazole, fluconazole, Ombitasvir, Dasabuvir, Carbamazepine, Enzalutamide, Fosphenytoin, Mitotane, Phenobarbital, Phenytoin, Rifampicin
3. Cardiogenic shock.
4. History of symptomatic bradycardia.
5. Renal insufficiency:eGFR<30 ml/min/1.73m2
6. Pregnancy
7. Heart failure due to congenital heart
8. Severe hypotension(<90/50mmHg)