Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Cos




Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Integration > Indefinite integration > Involving functions of the direct function and trigonometric functions > Involving algebraic functions of the direct function and trigonometric functions > Involving algebraic functions of sin > Involving (a+b sin(e z)+c cos(e z))beta





http://functions.wolfram.com/01.07.21.2387.01









  


  








Input Form





Integrate[Sin[e z]/Sqrt[a + b Sin[e z] + c Cos[e z]], z] == -((2 c^2 (a + c Cos[e z] + b Sin[e z]))/(b^2 + c^2) - (b Sin[e z - ArcTan[b/c]])/Sqrt[1 + b^2/c^2] + (b AppellF1[-(1/2), -(1/2), -(1/2), 1/2, (a + c Cos[e z] + b Sin[e z])/ (a + Sqrt[1 + b^2/c^2] c), (a + c Cos[e z] + b Sin[e z])/ (a - Sqrt[1 + b^2/c^2] c)] Sin[e z - ArcTan[b/c]])/ (Sqrt[1 + b^2/c^2] Sqrt[(Sqrt[1 + b^2/c^2] c - c Cos[e z] - b Sin[e z])/ (a + Sqrt[1 + b^2/c^2] c)] Sqrt[(Sqrt[1 + b^2/c^2] c + c Cos[e z] + b Sin[e z])/(-a + Sqrt[1 + b^2/c^2] c)]))/ (c e Sqrt[a + c Cos[e z] + b Sin[e z]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "-", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]]], ")"]]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["c", " ", "e", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mfrac> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <ci> b </ci> <apply> <ci> AppellF1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <ci> e </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cos /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sin /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["e_", " ", "z_"]], "]"]], SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sin", "[", RowBox[List["e_", " ", "z_"]], "]"]]]], "+", RowBox[List["c_", " ", RowBox[List["Cos", "[", RowBox[List["e_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["e", " ", "z"]], "-", RowBox[List["ArcTan", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "-", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]], "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]]]]]], RowBox[List["c", " ", "e", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cos", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2002-12-18