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 Cosh

 http://functions.wolfram.com/01.20.21.4067.01

 Input Form

 Integrate[Sinh[b Sqrt[z]] Cosh[c Sqrt[z]]^v, z] == (1/b^2) (2^(1 - v) Binomial[v, v/2] (1 - Mod[v, 2]) (b Sqrt[z] Cosh[b Sqrt[z]] - Sinh[b Sqrt[z]])) - 2^(1 - v) Sum[Binomial[v, s] (((-b + 2 c s - c v) Sqrt[z] Cosh[(-b + 2 c s - c v) Sqrt[z]] - Sinh[(-b + 2 c s - c v) Sqrt[z]])/(-b + 2 c s - c v)^2 + ((-b - 2 c s + c v) Sqrt[z] Cosh[(-b - 2 c s + c v) Sqrt[z]] - Sinh[(-b - 2 c s + c v) Sqrt[z]])/(-b - 2 c s + c v)^2), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["b", "2"]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SqrtBox["z"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]], ")"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh ( b z ) cosh v ( c z ) z 2 1 - v ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) ( b z cosh ( b z ) - sinh ( b z ) ) b 2 - 2 1 - v s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b - 2 c s + c v ) z cosh ( ( - b - 2 c s + c v ) z ) - sinh ( ( - b - 2 c s + c v ) z ) ( - b - 2 c s + c v ) 2 + 1 ( - b + 2 c s - c v ) 2 ( ( - b + 2 c s - c v ) z cosh ( ( - b + 2 c s - c v ) z ) - sinh ( ( - b + 2 c s - c v ) z ) ) ) /; m + v + Condition z b z 1 2 c z 1 2 v 2 1 -1 v Binomial v v 2 -1 1 -1 \$CellContext`v 2 b z 1 2 b z 1 2 -1 b z 1 2 b 2 -1 -1 2 1 -1 v s 0 v -1 2 -1 Binomial v s -1 b -1 2 c s c v z 1 2 -1 b -1 2 c s c v z 1 2 -1 -1 b -1 2 c s c v z 1 2 -1 b -1 2 c s c v 2 -1 1 -1 b 2 c s -1 c v 2 -1 -1 b 2 c s -1 c v z 1 2 -1 b 2 c s -1 c v z 1 2 -1 -1 b 2 c s -1 c v z 1 2 m SuperPlus v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SqrtBox["z"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]], ")"]]]], SuperscriptBox["b", "2"]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], "2"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18