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Cosh






Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Involving ((a+b cosh(c z))nu)beta





http://functions.wolfram.com/01.20.21.2310.01









  


  










Input Form





Integrate[((a + b Cosh[c z])^\[Nu])^\[Beta], z] == (1/(b c (1 + \[Beta] \[Nu]))) (AppellF1[1 + \[Beta] \[Nu], 1/2, 1/2, 2 + \[Beta] \[Nu], (a + b Cosh[c z])/(a + b), (a + b Cosh[c z])/(a - b)] Sqrt[(b (1 + Cosh[c z]))/(-a + b)] Sqrt[(b - b Cosh[c z])/(a + b)] (a + b Cosh[c z]) ((a + b Cosh[c z])^\[Nu])^\[Beta] Csch[c z])










Standard Form





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MathML Form







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</mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mrow> <mi> &#946; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <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> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <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> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> b </mi> <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> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <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> </mrow> </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> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </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> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> &#957; </ci> </apply> <ci> &#946; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> b </ci> <ci> c </ci> <apply> <plus /> <apply> <times /> <ci> &#946; </ci> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> AppellF1 </ci> <apply> <plus /> <apply> <times /> <ci> &#946; </ci> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> &#946; </ci> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <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> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> &#957; </ci> </apply> <ci> &#946; </ci> </apply> <apply> <csch /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "\[Nu]_"], ")"]], "\[Beta]_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["\[Beta]", " ", "\[Nu]"]]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["2", "+", RowBox[List["\[Beta]", " ", "\[Nu]"]]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]], ",", FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", "b"]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "\[Nu]"], ")"]], "\[Beta]"], " ", RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["b", " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Beta]", " ", "\[Nu]"]]]], ")"]]]]]]]]]










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





2002-12-18