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
 











Csc




Mathematica Notation

Traditional Notation









Elementary Functions > Csc[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Involving ((a+b csc2(c z))n)beta





http://functions.wolfram.com/01.10.21.0107.01









  


  








Input Form





Integrate[Csc[c z] ((a + b Csc[c z]^2)^n)^\[Beta], z] == (-(1/c)) ((AppellF1[1/2, 1/2, (-n) \[Beta], 3/2, Csc[c z]^2, -((b Csc[c z]^2)/a)] Sqrt[-Cot[c z]^2] ((a + b Csc[c z]^2)^n)^\[Beta] Sec[c z])/(1 + (b Csc[c z]^2)/a)^(n \[Beta]))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "n"], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "c"]]], RowBox[List["(", RowBox[List[RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List[RowBox[List["-", "n"]], " ", "\[Beta]"]], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "a"]]]]], "]"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "n"], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "a"]]], ")"]], RowBox[List[RowBox[List["-", "n"]], " ", "\[Beta]"]]], " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "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> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mo> - </mo> <mrow> <msup> <mi> cot </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> c </mi> </mfrac> </mrow> <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> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> n </ci> </apply> <ci> &#946; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <cot /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> </apply> <ci> n </ci> </apply> <ci> &#946; </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> &#946; </ci> </apply> </apply> <apply> <sec /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> AppellF1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> &#946; </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Csc", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "n_"], ")"]], "\[Beta]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List[RowBox[List["-", "n"]], " ", "\[Beta]"]], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "a"]]]]], "]"]], " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "n"], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], "a"]]], ")"]], RowBox[List[RowBox[List["-", "n"]], " ", "\[Beta]"]]], " ", RowBox[List["Sec", "[", RowBox[List["c", " ", "z"]], "]"]]]], "c"]]]]]]]


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





2002-12-18