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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving powers of the direct function > Involving powers of sin > Involving sinhv(a z)





http://functions.wolfram.com/01.19.21.3418.01









  


  










Input Form





Integrate[Sinh[a z]^(2 n), z] == ((-1)^n a z Pochhammer[1/2, n])/(a n!) - ((-1)^n Pochhammer[1/2, n] ArcSinh[Sinh[a z]] Cosh[a z])/ (a n! Sqrt[Cosh[a z]^2]) + ((Cosh[a z] Sinh[a z]^(1 + 2 n))/ (a (1 + 2 n))) Hypergeometric2F1[1, 1 + n, 3/2 + n, -Sinh[a z]^2] /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> a </ci> <ci> z </ci> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <factorial /> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> n </ci> </apply> <apply> <arcsinh /> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <apply> <cosh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["a_", " ", "z_"]], "]"]], RowBox[List["2", " ", "n_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "a", " ", "z", " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]]], RowBox[List["a", " ", RowBox[List["n", "!"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], " ", RowBox[List["ArcSinh", "[", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], RowBox[List["a", " ", RowBox[List["n", "!"]], " ", SqrtBox[SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], "2"]]]]], "]"]]]], RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02