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 Cosh

 http://functions.wolfram.com/01.20.21.4069.01

 Input Form

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

 MathML Form

 sinh ( b z 2 + e ) cosh v ( c z 2 ) z 2 - v - 2 - e π ( 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]]]]] ( - b erfi ( - b z ) + b 2 e erfi ( b z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) b + 2 - v - 2 π 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]]]]] ( - e - b + 2 c s - c v erfi ( 2 c ( 2 s - v ) z - 2 b z 2 - b + 2 c s - c v ) b + c ( v - 2 s ) + 1 b + 2 c s - c v ( - e ( - b - 2 c s + c v erfi ( - 2 b z - 2 c ( 2 s - v ) z 2 - b - 2 c s + c v ) + 2 e b + 2 c s - c v erfi ( 2 b z + 2 c ( 2 s - v ) z 2 b + 2 c s - c v ) ) ) + e erfi ( 2 b z - 2 c ( 2 s - v ) z 2 b + c ( v - 2 s ) ) b + c ( v - 2 s ) ) /; v + Condition z b z 2 e c z 2 v 2 -1 v -2 -1 e 1 2 Binomial v v 2 -1 -1 b 1 2 Erfi -1 b 1 2 z b 1 2 2 e Erfi b 1 2 z 1 -1 \$CellContext`v 2 b -1 2 -1 v -2 1 2 s 0 v -1 2 -1 Binomial v s -1 e -1 b 2 c s -1 c v 1 2 Erfi 2 c 2 s -1 v z -1 2 b z 2 -1 b 2 c s -1 c v 1 2 -1 b c v -1 2 s -1 1 b 2 c s -1 c v -1 -1 e -1 b -1 2 c s c v 1 2 Erfi -2 b z -1 2 c 2 s -1 v z 2 -1 b -1 2 c s c v 1 2 -1 2 e b 2 c s -1 c v 1 2 Erfi 2 b z 2 c 2 s -1 v z 2 b 2 c s -1 c v 1 2 -1 e Erfi 2 b z -1 2 c 2 s -1 v z 2 b c v -1 2 s 1 2 -1 b c v -1 2 s 1 2 -1 v SuperPlus [/itex]

 Rule Form

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

 2002-12-18