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http://functions.wolfram.com/01.07.21.0157.01
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Integrate[(z Cos[c z])/(a z^2 + b), z] ==
(1/(2 a)) (Cosh[(Sqrt[b] c)/Sqrt[a]] CosIntegral[
c (-((I Sqrt[b])/Sqrt[a]) + z)] + Cosh[(Sqrt[b] c)/Sqrt[a]]
CosIntegral[c ((I Sqrt[b])/Sqrt[a] + z)] + I Sinh[(Sqrt[b] c)/Sqrt[a]]
(SinIntegral[c ((I Sqrt[b])/Sqrt[a] + z)] +
SinIntegral[(I Sqrt[b] c)/Sqrt[a] - c z]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", "b"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "a"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]]]], "+", "z"]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["SinIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]], "]"]], "+", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "-", RowBox[List["c", " ", "z"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Ci </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> b </mi> </msqrt> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Ci </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> b </mi> </msqrt> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Si </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> b </mi> </msqrt> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> Si </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </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> z </ci> </bvar> <apply> <times /> <ci> z </ci> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> CosIntegral </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <cosh /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> CosIntegral </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <sinh /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> SinIntegral </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> SinIntegral </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["z_", " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", "b_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]]]], "+", "z"]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List[SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["SinIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"]]], SqrtBox["a"]], "+", "z"]], ")"]]]], "]"]], "+", RowBox[List["SinIntegral", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["b"], " ", "c"]], SqrtBox["a"]], "-", RowBox[List["c", " ", "z"]]]], "]"]]]], ")"]]]]]], RowBox[List["2", " ", "a"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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