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CoshIntegral






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > CoshIntegral[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions and a power function > Involving cosh and power





http://functions.wolfram.com/06.40.21.0042.01









  


  










Input Form





Integrate[z^3 Cosh[b z] CoshIntegral[a z], z] == (1/(2 b^4)) (2 CoshIntegral[a z] (-3 (2 + b^2 z^2) Cosh[b z] + b z (6 + b^2 z^2) Sinh[b z]) + (1/(a^2 - b^2)^3) (3 (a^2 - b^2)^3 (ExpIntegralEi[(a - b) z] + ExpIntegralEi[(-a + b) z] + ExpIntegralEi[(-(a + b)) z] + ExpIntegralEi[(a + b) z]) - 2 b^2 Cosh[a z] ((-3 a^4 + 6 a^2 b^2 - 11 b^4 - b^2 (a^2 - b^2)^2 z^2) Cosh[b z] + b (a^2 - 5 b^2) (a^2 - b^2) z Sinh[b z]) + 2 a b Sinh[a z] (b (-a^2 + b^2) (-3 a^2 + 7 b^2) z Cosh[b z] + (-2 (3 a^4 - 8 a^2 b^2 + 9 b^4) - b^2 (a^2 - b^2)^2 z^2) Sinh[b z])))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["CoshIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["b", "4"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["CoshIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", "z", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "3"]], RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]], "]"]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], " ", "z"]], "]"]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["a", "4"]]], "+", RowBox[List["6", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["11", " ", SuperscriptBox["b", "4"]]], "-", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["5", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", "z", " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", "a", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["a", "2"]]], "+", RowBox[List["7", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["8", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["9", " ", SuperscriptBox["b", "4"]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]










MathML Form







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<apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> z </ci> <apply> <sinh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <ci> z </ci> <apply> <cosh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <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> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.