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
 











Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving rational functions > Involving (a z+b)-n





http://functions.wolfram.com/01.19.21.0139.01









  


  










Input Form





Integrate[Sinh[c z]/(a z + b)^4, z] == (-(1/(6 a^4))) ((a^2 c Cosh[c z])/(b + a z)^2 + (a (b^2 c^2 + 2 a b c^2 z + a^2 (2 + c^2 z^2)) Sinh[c z])/(b + a z)^3 - c^3 (Cosh[(b c)/a] CoshIntegral[c (b/a + z)] - Sinh[(b c)/a] SinhIntegral[c (b/a + z)]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], ")"]], "4"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["6", " ", SuperscriptBox["a", "4"]]]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "c", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["2", " ", "a", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "3"]], "-", RowBox[List[SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "z"]], ")"]]]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "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> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Chi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Shi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> c </mi> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <ci> b </ci> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> CoshIntegral </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <sinh /> <apply> <times /> <ci> b </ci> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> SinhIntegral </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> c </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <ci> a </ci> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], ")"]], "4"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "c", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["2", " ", "a", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], "3"]], "-", RowBox[List[SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "z"]], ")"]]]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "z"]], ")"]]]], "]"]]]]]], ")"]]]]]], RowBox[List["6", " ", SuperscriptBox["a", "4"]]]]]]]]]]










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





2002-12-18