By analyzing <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>7.33</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:msup><a:mi>fb</a:mi><a:mrow><a:mo>−</a:mo><a:mn>1</a:mn></a:mrow></a:msup></a:math> of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>e</c:mi><c:mo>+</c:mo></c:msup><c:msup><c:mi>e</c:mi><c:mo>−</c:mo></c:msup></c:math> annihilation data collected at center-of-mass energies between 4.128 and 4.226 GeV with the BESIII detector, we report the observation of the semileptonic decay <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msubsup><e:mi>D</e:mi><e:mi>s</e:mi><e:mo>+</e:mo></e:msubsup><e:mo stretchy="false">→</e:mo><e:msup><e:mi>η</e:mi><e:mo>′</e:mo></e:msup><e:msup><e:mi>μ</e:mi><e:mo>+</e:mo></e:msup><e:msub><e:mi>ν</e:mi><e:mi>μ</e:mi></e:msub></e:math>, with a statistical significance larger than <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mrow><h:mn>10</h:mn><h:mi>σ</h:mi></h:mrow></h:math>, and the measurements of the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msubsup><j:mi>D</j:mi><j:mi>s</j:mi><j:mo>+</j:mo></j:msubsup><j:mo stretchy="false">→</j:mo><j:mi>η</j:mi><j:msup><j:mi>μ</j:mi><j:mo>+</j:mo></j:msup><j:msub><j:mi>ν</j:mi><j:mi>μ</j:mi></j:msub></j:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mi>D</m:mi><m:mi>s</m:mi><m:mo>+</m:mo></m:msubsup><m:mo stretchy="false">→</m:mo><m:msup><m:mi>η</m:mi><m:mo>′</m:mo></m:msup><m:msup><m:mi>μ</m:mi><m:mo>+</m:mo></m:msup><m:msub><m:mi>ν</m:mi><m:mi>μ</m:mi></m:msub></m:math> decay dynamics for the first time. The branching fractions of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msubsup><p:mi>D</p:mi><p:mi>s</p:mi><p:mo>+</p:mo></p:msubsup><p:mo stretchy="false">→</p:mo><p:mi>η</p:mi><p:msup><p:mi>μ</p:mi><p:mo>+</p:mo></p:msup><p:msub><p:mi>ν</p:mi><p:mi>μ</p:mi></p:msub></p:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msubsup><s:mi>D</s:mi><s:mi>s</s:mi><s:mo>+</s:mo></s:msubsup><s:mo stretchy="false">→</s:mo><s:msup><s:mi>η</s:mi><s:mo>′</s:mo></s:msup><s:msup><s:mi>μ</s:mi><s:mo>+</s:mo></s:msup><s:msub><s:mi>ν</s:mi><s:mi>μ</s:mi></s:msub></s:math> are determined to be <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mo stretchy="false">(</v:mo><v:mn>2.235</v:mn><v:mo>±</v:mo><v:mn>0.05</v:mn><v:msub><v:mn>1</v:mn><v:mrow><v:mi>stat</v:mi></v:mrow></v:msub><v:mo>±</v:mo><v:mn>0.05</v:mn><v:msub><v:mn>2</v:mn><v:mrow><v:mi>syst</v:mi></v:mrow></v:msub><v:mo stretchy="false">)</v:mo><v:mo>%</v:mo></v:math> and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mo stretchy="false">(</z:mo><z:mn>0.801</z:mn><z:mo>±</z:mo><z:mn>0.05</z:mn><z:msub><z:mn>5</z:mn><z:mrow><z:mi>stat</z:mi></z:mrow></z:msub><z:mo>±</z:mo><z:mn>0.02</z:mn><z:msub><z:mn>8</z:mn><z:mrow><z:mi>syst</z:mi></z:mrow></z:msub><z:mo stretchy="false">)</z:mo><z:mo>%</z:mo></z:math>, respectively, with precision improved by factors of 6.0 and 6.6 compared to the previous best measurements. Combined with the results for the decays <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:msubsup><db:mi>D</db:mi><db:mi>s</db:mi><db:mo>+</db:mo></db:msubsup><db:mo stretchy="false">→</db:mo><db:mi>η</db:mi><db:msup><db:mi>e</db:mi><db:mo>+</db:mo></db:msup><db:msub><db:mi>ν</db:mi><db:mi>e</db:mi></db:msub></db:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msubsup><gb:mi>D</gb:mi><gb:mi>s</gb:mi><gb:mo>+</gb:mo></gb:msubsup><gb:mo stretchy="false">→</gb:mo><gb:msup><gb:mi>η</gb:mi><gb:mo>′</gb:mo></gb:msup><gb:msup><gb:mi>e</gb:mi><gb:mo>+</gb:mo></gb:msup><gb:msub><gb:mi>ν</gb:mi><gb:mi>e</gb:mi></gb:msub></gb:math>, the ratios of the decay widths are examined both inclusively and in several <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:msup><jb:mo>ℓ</jb:mo><jb:mo>+</jb:mo></jb:msup><jb:msub><jb:mi>ν</jb:mi><jb:mo>ℓ</jb:mo></jb:msub></jb:math> four-momentum transfer ranges. No evidence for lepton flavor universality violation is found within the current statistics. The products of the hadronic form factors <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:msubsup><lb:mi>f</lb:mi><lb:mrow><lb:mo>+</lb:mo><lb:mo>,</lb:mo><lb:mn>0</lb:mn></lb:mrow><lb:msup><lb:mi>η</lb:mi><lb:mrow><lb:mo stretchy="false">(</lb:mo><lb:mo>′</lb:mo><lb:mo stretchy="false">)</lb:mo></lb:mrow></lb:msup></lb:msubsup><lb:mo stretchy="false">(</lb:mo><lb:mn>0</lb:mn><lb:mo stretchy="false">)</lb:mo></lb:math> and the <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:mi>c</rb:mi><rb:mo stretchy="false">→</rb:mo><rb:mi>s</rb:mi></rb:math> Cabibbo-Kobayashi-Maskawa matrix element <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mo stretchy="false">|</ub:mo><ub:msub><ub:mi>V</ub:mi><ub:mrow><ub:mi>c</ub:mi><ub:mi>s</ub:mi></ub:mrow></ub:msub><ub:mo stretchy="false">|</ub:mo></ub:math> are determined. The results based on the two-parameter series expansion are <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msubsup><yb:mi>f</yb:mi><yb:mrow><yb:mo>+</yb:mo><yb:mo>,</yb:mo><yb:mn>0</yb:mn></yb:mrow><yb:mi>η</yb:mi></yb:msubsup><yb:mo stretchy="false">(</yb:mo><yb:mn>0</yb:mn><yb:mo stretchy="false">)</yb:mo><yb:mo stretchy="false">|</yb:mo><yb:msub><yb:mi>V</yb:mi><yb:mrow><yb:mi>c</yb:mi><yb:mi>s</yb:mi></yb:mrow></yb:msub><yb:mo stretchy="false">|</yb:mo><yb:mo>=</yb:mo><yb:mn>0.452</yb:mn><yb:mo>±</yb:mo><yb:mn>0.01</yb:mn><yb:msub><yb:mn>0</yb:mn><yb:mrow><yb:mi>stat</yb:mi></yb:mrow></yb:msub><yb:mo>±</yb:mo><yb:mn>0.00</yb:mn><yb:msub><yb:mn>7</yb:mn><yb:mrow><yb:mi>syst</yb:mi></yb:mrow></yb:msub></yb:math> and <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msubsup><ec:mi>f</ec:mi><ec:mrow><ec:mo>+</ec:mo><ec:mo>,</ec:mo><ec:mn>0</ec:mn></ec:mrow><ec:msup><ec:mi>η</ec:mi><ec:mo>′</ec:mo></ec:msup></ec:msubsup><ec:mo stretchy="false">(</ec:mo><ec:mn>0</ec:mn><ec:mo stretchy="false">)</ec:mo><ec:mo stretchy="false">|</ec:mo><ec:msub><ec:mi>V</ec:mi><ec:mrow><ec:mi>c</ec:mi><ec:mi>s</ec:mi></ec:mrow></ec:msub><ec:mo stretchy="false">|</ec:mo><ec:mo>=</ec:mo><ec:mn>0.504</ec:mn><ec:mo>±</ec:mo><ec:mn>0.03</ec:mn><ec:msub><ec:mn>7</ec:mn><ec:mrow><ec:mi>stat</ec:mi></ec:mrow></ec:msub><ec:mo>±</ec:mo><ec:mn>0.01</ec:mn><ec:msub><ec:mn>2</ec:mn><ec:mrow><ec:mi>syst</ec:mi></ec:mrow></ec:msub></ec:math>, which help to constrain present models on <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:msubsup><kc:mi>f</kc:mi><kc:mrow><kc:mo>+</kc:mo><kc:mo>,</kc:mo><kc:mn>0</kc:mn></kc:mrow><kc:msup><kc:mi>η</kc:mi><kc:mrow><kc:mo stretchy="false">(</kc:mo><kc:mo>′</kc:mo><kc:mo stretchy="false">)</kc:mo></kc:mrow></kc:msup></kc:msubsup><kc:mo stretchy="false">(</kc:mo><kc:mn>0</kc:mn><kc:mo stretchy="false">)</kc:mo></kc:math>. The forward-backward asymmetries are determined to be <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:mo stretchy="false">⟨</qc:mo><qc:msubsup><qc:mi>A</qc:mi><qc:mi>FB</qc:mi><qc:mi>η</qc:mi></qc:msubsup><qc:mo stretchy="false">⟩</qc:mo><qc:mo>=</qc:mo><qc:mo>−</qc:mo><qc:mn>0.059</qc:mn><qc:mo>±</qc:mo><qc:mn>0.03</qc:mn><qc:msub><qc:mn>1</qc:mn><qc:mrow><qc:mi>stat</qc:mi></qc:mrow></qc:msub><qc:mo>±</qc:mo><qc:mn>0.00</qc:mn><qc:msub><qc:mn>5</qc:mn><qc:mrow><qc:mi>syst</qc:mi></qc:mrow></qc:msub></qc:math> and <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" display="inline"><uc:mo stretchy="false">⟨</uc:mo><uc:msubsup><uc:mi>A</uc:mi><uc:mi>FB</uc:mi><uc:msup><uc:mi>η</uc:mi><uc:mo>′</uc:mo></uc:msup></uc:msubsup><uc:mo stretchy="false">⟩</uc:mo><uc:mo>=</uc:mo><uc:mo>−</uc:mo><uc:mn>0.064</uc:mn><uc:mo>±</uc:mo><uc:mn>0.07</uc:mn><uc:msub><uc:mn>9</uc:mn><uc:mrow><uc:mi>stat</uc:mi></uc:mrow></uc:msub><uc:mo>±</uc:mo><uc:mn>0.00</uc:mn><uc:msub><uc:mn>6</uc:mn><uc:mrow><uc:mi>syst</uc:mi></uc:mrow></uc:msub></uc:math> for the first time, which are consistent with the theoretical calculation.
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