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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 sinh > Involving sinhm(c z)coshm(c z)(a+b cosh(2c z))beta





http://functions.wolfram.com/01.20.21.4359.01









  


  










Input Form





Integrate[(Sinh[c z]^3 Cosh[c z]^3)/Sqrt[a + b Cosh[2 c z]], z] == (32 a^3 - 62 a b^2 + b (16 a^2 - 51 b^2) Cosh[2 c z] - 32 a^3 Sqrt[1 + (b Cosh[2 c z])/a] + 9 a b^2 Sqrt[1 + (b Cosh[2 c z])/a] - 2 a b^2 Cosh[4 c z] + 3 b^3 Cosh[6 c z])/ (480 b^3 c Sqrt[a + b Cosh[2 c z]])










Standard Form





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MathML Form







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</mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 480 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -32 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 62 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 51 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 480 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <ci> c </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </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





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "3"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "3"]]], SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["32", " ", SuperscriptBox["a", "3"]]], "-", RowBox[List["62", " ", "a", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["16", " ", SuperscriptBox["a", "2"]]], "-", RowBox[List["51", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "-", RowBox[List["32", " ", SuperscriptBox["a", "3"], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "a"]]]]]], "+", RowBox[List["9", " ", "a", " ", SuperscriptBox["b", "2"], " ", SqrtBox[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], "a"]]]]]], "-", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["4", " ", "c", " ", "z"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox["b", "3"], " ", RowBox[List["Cosh", "[", RowBox[List["6", " ", "c", " ", "z"]], "]"]]]]]], RowBox[List["480", " ", SuperscriptBox["b", "3"], " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]]]]]]










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





2002-12-18