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Cosh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.20.21.4410.01









  


  








Input Form





Integrate[Cosh[e z]/Sqrt[a + b Sinh[e z] + c Cosh[e z]], z] == ((2 (a + c Cosh[e z] + b Sinh[e z]))/b + (c ((-(b^2 - 2 c^2)) Cosh[e z] + c (2 a + b Sinh[e z])))/(b^3 - b c^2) - (AppellF1[-(1/2), -(1/2), -(1/2), 1/2, (a + c Cosh[e z] + b Sinh[e z])/ (a + Sqrt[1 - b^2/c^2] c), (a + c Cosh[e z] + b Sinh[e z])/ (a - Sqrt[1 - b^2/c^2] c)] Sinh[e z + ArcTanh[b/c]])/ (Sqrt[1 - b^2/c^2] Sqrt[(Sqrt[1 - b^2/c^2] c (1 + Cosh[e z + ArcTanh[b/c]]))/(-a + Sqrt[1 - b^2/c^2] c)] Sqrt[(Sqrt[1 - b^2/c^2] c - c Cosh[e z] - b Sinh[e z])/ (a + Sqrt[1 - b^2/c^2] c)]))/(e Sqrt[a + c Cosh[e z] + b Sinh[e z]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]], "b"], "+", FractionBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["2", " ", SuperscriptBox["c", "2"]]]]], ")"]]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "3"], "-", RowBox[List["b", " ", SuperscriptBox["c", "2"]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[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["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", 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["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["e", " ", "z"]], "+", RowBox[List["ArcTanh", "[", FractionBox["b", "c"], "]"]]]], "]"]]]], ")"]]]], RowBox[List[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["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["e", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]








MathML Form







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</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> cosh </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> sinh </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> <mn> 1 </mn> <mo> - </mo> <mfrac> 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<plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> <apply> <plus /> <apply> <cosh /> <apply> <plus /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> <apply> <arctanh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </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> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </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> <sinh /> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </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> <cn type='integer'> -1 </cn> </apply> </apply> <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> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </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> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> e </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> c </ci> <apply> <cosh /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sinh /> <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





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





2002-12-18