We measure the inclusive forward-backward asymmetry of the charged-lepton pseudorapidities from top-quark pairs produced in proton-antiproton collisions, and decaying to final states that contain two charged leptons (electrons or muons), using data collected with the Collider Detector at Fermilab. With an integrated luminosity of 9.1 $\rm{fb}^{-1}$, the leptonic forward-backward asymmetry, $A_{\text{FB}}^{\ell}$, is measured to be $0.072 \pm 0.060$ and the leptonic pair forward-backward asymmetry, $A_{\text{FB}}^{\ell\ell}$, is measured to be $0.076 \pm 0.082$, compared with the standard model predictions of $A_{\text{FB}}^{\ell} = 0.038 \pm 0.003$ and $A_{\text{FB}}^{\ell\ell} = 0.048 \pm 0.004$, respectively. Additionally, we combine the $A_{\text{FB}}^{\ell}$ result with a previous determination from a final state with a single lepton and hadronic jets and obtain $A_{\text{FB}}^{\ell} = 0.090^{+0.028}_{-0.026}$.
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