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http://functions.wolfram.com/01.04.21.0038.01
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Integrate[Log[1 + t] Log[1 + 1/t^2], {t, 0, Infinity}] ==
2 Catalan - Pi + (5 Pi^2)/24 + (1/2) Pi Log[2]
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Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "t"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["1", SuperscriptBox["t", "2"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List["2", " ", "Catalan"]], "-", "\[Pi]", "+", FractionBox[RowBox[List["5", " ", SuperscriptBox["\[Pi]", "2"]]], "24"], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Log", "[", "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> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mn> 24 </mn> </mfrac> <mo> - </mo> <mi> π </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", Function[Catalan]] </annotation> </semantics> </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> <infinity /> </uplimit> <apply> <times /> <apply> <ln /> <apply> <plus /> <ci> t </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 24 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> Catalan </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "t_"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox["1", SuperscriptBox["t_", "2"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", "Catalan"]], "-", "\[Pi]", "+", FractionBox[RowBox[List["5", " ", SuperscriptBox["\[Pi]", "2"]]], "24"], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Log", "[", "2", "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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