Integrate[t^\[Alpha] PolyLog[2, t], {t, 0, 1}] == (1/(6 \[Alpha] (1 + \[Alpha])^3)) (-6 + \[Alpha] (-12 + Pi^2 (1 + \[Alpha])^2) - 6 \[Alpha] (1 + \[Alpha]) (EulerGamma + PolyGamma[\[Alpha]])) /; Re[\[Alpha]] > -2

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <msup> <mi> t </mi> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mi> &#945; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 12 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> &#945; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> &#945; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <ci> &#945; </ci> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> &#945; </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -12 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <ci> &#945; </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> PolyGamma </ci> <ci> &#968; </ci> </apply> <ci> &#945; </ci> </apply> <eulergamma /> </apply> </apply> </apply> <cn type='integer'> -6 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> &#945; </ci> <apply> <power /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> &#945; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29