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 Cosh

 http://functions.wolfram.com/01.20.21.4046.01

 Input Form

 Integrate[Sinh[b Sqrt[z]] Cosh[c z]^v, z] == 2^(-2 - v) (-((1/b^2) ((4 (1 + E^(2 b Sqrt[z]) (-1 + b Sqrt[z]) + b Sqrt[z]) Binomial[v, v/2] (-1 + Mod[v, 2]))/E^(b Sqrt[z]))) + Sum[Binomial[v, s] ((8 Sinh[b Sqrt[z]] Sinh[c (2 s - v) z])/ (c (2 s - v)) + (b E^(b^2/(-8 c s + 4 c v)) Sqrt[Pi] Erfi[(-b + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (2 s - v)])])/ (c (2 s - v))^(3/2) - (b E^(b^2/(8 c s - 4 c v)) Sqrt[Pi] Erfi[(-b + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (-2 s + v)])])/ (c (-2 s + v))^(3/2) + (b E^(b^2/(-8 c s + 4 c v)) Sqrt[Pi] Erfi[(b + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (2 s - v)])])/ (c (2 s - v))^(3/2) - (b E^(b^2/(8 c s - 4 c v)) Sqrt[Pi] Erfi[(b + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (-2 s + v)])])/ (c (-2 s + v))^(3/2)), {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", " ", "z"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", SuperscriptBox["b", "2"]], RowBox[List["(", RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "b"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", SqrtBox["z"]]]]], ")"]]]], "+", RowBox[List["b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", 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RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List[RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh ( b z ) cosh v ( c z ) z 2 - v - 2 ( s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( b b 2 4 c v - 8 c s π erfi ( 2 c ( v - 2 s ) z - b 2 c ( 2 s - v ) ) ( c ( 2 s - v ) ) 3 / 2 + b b 2 4 c v - 8 c s π erfi ( b + 2 c ( v - 2 s ) z 2 c ( 2 s - v ) ) ( c ( 2 s - v ) ) 3 / 2 + 8 sinh ( b z ) sinh ( c ( 2 s - v ) z ) c ( 2 s - v ) - b b 2 8 c s - 4 c v π erfi ( 2 c ( v - 2 s ) z - b 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 3 / 2 - b b 2 8 c s - 4 c v π erfi ( b + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 3 / 2 ) - 4 - b z ( z b + 2 b z ( b z - 1 ) + 1 ) ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( v mod 2 \$CellContext`v 2 - 1 ) b 2 ) /; v + Condition z b z 1 2 c z v 2 -1 v -2 s 0 v -1 2 -1 Binomial v s b b 2 4 c v -1 8 c s -1 1 2 Erfi 2 c v -1 2 s z 1 2 -1 b 2 c 2 s -1 v 1 2 -1 c 2 s -1 v 3 2 -1 b b 2 4 c v -1 8 c s -1 1 2 Erfi b 2 c v -1 2 s z 1 2 2 c 2 s -1 v 1 2 -1 c 2 s -1 v 3 2 -1 8 b z 1 2 c 2 s -1 v z c 2 s -1 v -1 -1 b b 2 8 c s -1 4 c v -1 1 2 Erfi 2 c v -1 2 s z 1 2 -1 b 2 c v -1 2 s 1 2 -1 c v -1 2 s 3 2 -1 -1 b b 2 8 c s -1 4 c v -1 1 2 Erfi b 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c v -1 2 s 3 2 -1 -1 4 -1 b z 1 2 z 1 2 b 2 b z 1 2 b z 1 2 -1 1 Binomial v v 2 -1 \$CellContext`v 2 -1 b 2 -1 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_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "b"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", SqrtBox["z"]]]]], ")"]]]], "+", RowBox[List["b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18