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http://functions.wolfram.com/01.11.21.0017.01
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Integrate[t^2 Sec[t], {t, 0, Pi/4}] ==
(1/64) (Pi (4 Pi (-2 I Pi + Log[1 - (-1)^(3/4)] - Log[1 + (-1)^(3/4)]) +
(-1)^(1/4) (PolyGamma[1, 1/8] - I PolyGamma[1, 3/8] -
PolyGamma[1, 5/8] + I PolyGamma[1, 7/8])) +
128 (PolyLog[3, (-1)^(3/4)] - PolyLog[3, -(-1)^(3/4)]))
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Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox["\[Pi]", "4"]], RowBox[List[SuperscriptBox["t", "2"], " ", RowBox[List["Sec", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "64"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox["1", "8"]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox["3", "8"]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox["5", "8"]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox["7", "8"]]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["128", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["3", ",", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]
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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> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> sec </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 64 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mroot> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 8 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 5 </mn> <mn> 8 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 7 </mn> <mn> 8 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 128 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> <apply> <sec /> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 64 </cn> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <cn type='rational'> 1 <sep /> 8 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <cn type='rational'> 3 <sep /> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <cn type='rational'> 5 <sep /> 8 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <cn type='rational'> 7 <sep /> 8 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 128 </cn> <apply> <plus /> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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