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 rational functions of the direct function > Involving (a+b csc2(z))-n





http://functions.wolfram.com/01.10.21.0091.01









  


  








Input Form





Integrate[1/(a + b Csc[z]^2)^2, z] == ((-a - 2 b + a Cos[2 z]) Csc[z]^4 ((Sqrt[b] (3 a + 2 b) ArcTan[(Sqrt[a + b] Tan[z])/Sqrt[b]] (a + 2 b - a Cos[2 z]))/(a + b)^(3/2) + 2 z (-a - 2 b + a Cos[2 z]) - (a b Sin[2 z])/(a + b)))/(8 a^2 (a + b Csc[z]^2)^2)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["Tan", "[", "z", "]"]]]], SqrtBox["b"]], "]"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "-", RowBox[List["a", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]], "-", FractionBox[RowBox[List["a", " ", "b", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List["a", "+", "b"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["8", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"]]]]], ")"]], "2"]]], ")"]]]]]]]]








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> <mn> 1 </mn> <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> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 4 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> b </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> a </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <csc /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </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'> 8 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <csc /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <csc /> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <tan /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <ci> a </ci> </apply> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> b </ci> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Csc", "[", "z_", "]"]], "2"]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox["b"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "a"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["Tan", "[", "z", "]"]]]], SqrtBox["b"]], "]"]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b"]], "-", RowBox[List["a", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]], "-", FractionBox[RowBox[List["a", " ", "b", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "z"]], "]"]]]], RowBox[List["a", "+", "b"]]]]], ")"]]]], RowBox[List["8", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"]]]]], ")"]], "2"]]]]]]]]


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





2002-12-18