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http://functions.wolfram.com/01.20.21.0711.01
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Integrate[Cos[b Sqrt[z] + d z] Cosh[f z + g], z] ==
(1/4) ((-(1/(-d - I f)^(3/2))) (b Sqrt[2 Pi] Cos[-(b^2/(4 (d + I f))) + I g]
FresnelC[(b + 2 (d + I f) Sqrt[z])/(Sqrt[-d - I f] Sqrt[2 Pi])] +
b Sqrt[2 Pi] FresnelS[(b + 2 (d + I f) Sqrt[z])/(Sqrt[-d - I f]
Sqrt[2 Pi])] Sin[-(b^2/(4 (d + I f))) + I g] +
2 Sqrt[-d - I f] Sin[I g + b Sqrt[z] + (d + I f) z]) +
(1/(d - I f)^(3/2)) ((-b) Sqrt[2 Pi] Cos[b^2/(4 (d - I f)) + I g]
FresnelC[(b + 2 (d - I f) Sqrt[z])/(Sqrt[d - I f] Sqrt[2 Pi])] -
b Sqrt[2 Pi] FresnelS[(b + 2 (d - I f) Sqrt[z])/(Sqrt[d - I f]
Sqrt[2 Pi])] Sin[b^2/(4 (d - I f)) + I g] +
2 Sqrt[d - I f] Sin[(-I) g + b Sqrt[z] + (d - I f) z]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["b", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["b", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List["b", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]], "+", RowBox[List["\[ImaginaryI]", " ", "g"]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "g"]], "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", "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> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> g </mi> <mo> + </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mrow> <msqrt> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mrow> <msqrt> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> g </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> f </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </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 /> <apply> <cos /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <cosh /> <apply> <plus /> <ci> g </ci> <apply> <times /> <ci> f </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> f </ci> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> f </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> g </ci> </apply> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> f </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> f </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn 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){kind=small}
