Hyperbolic cosine integral: Integration (formula 01.20.21.2312)

Cosh

$Mathematica Notation$

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/01.20.21.2312.01

Input Form

Integrate[Sqrt[(a + b Cosh[c z])^3], z] == (1/(3 c (a + b Cosh[c z])^2)) (Sqrt[(a + b Cosh[c z])^3] (-8 I a (a + b) Sqrt[(a + b Cosh[c z])/(a + b)] EllipticE[(I c z)/2, (2 b)/(a + b)] + 2 I (a^2 - b^2) Sqrt[(a + b Cosh[c z])/(a + b)] EllipticF[(I c z)/2, (2 b)/(a + b)] + 2 b (a + b Cosh[c z]) Sinh[c z]))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["3", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]]]], RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "2"], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "2"], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "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> ∫ </mo> <mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </msqrt> <mtext> </mtext> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ❘ </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ❘ </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </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> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <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> <cn type='integer'> 3 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> <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> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -8 </cn> <ci> a </ci> <apply> <power /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <imaginaryi /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <imaginaryi /> <apply> <ci> EllipticF </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <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> <sinh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "2"], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", "b"]]]], " ", RowBox[List["EllipticF", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "2"], ",", FractionBox[RowBox[List["2", " ", "b"]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List["3", " ", "c", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]]]]]]]]

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

2002-12-18