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 Cosh

 http://functions.wolfram.com/01.20.21.4490.01

 Input Form

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

 Standard Form

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 MathML Form

 z n sinh ( z b + e ) cosh v ( z c + g ) z 2 - v - 1 ( 2 b - 2 n - 2 - e ( 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]]]]] ( 2 e Γ ( 2 ( n + 1 ) , - b z ) - Γ ( 2 ( n + 1 ) , b z ) ) ( v mod 2 \$CellContext`v 2 - 1 ) - 2 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 + 2 g s - g v Γ ( 2 ( n + 1 ) , ( - b - 2 c s + c v ) z ) ( - b - 2 c s + c v ) - 2 ( n + 1 ) + e + g ( v - 2 s ) ( - b + 2 c s - c v ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - b + 2 c s - c v ) z ) - - e - 2 g s + g v ( b + 2 c s - c v ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( b + 2 c s - c v ) z ) - - e + 2 g s - g v ( b + c ( v - 2 s ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( b + c ( v - 2 s ) ) z ) ) ) /; n v + Condition z z n z 1 2 b e z 1 2 c g v 2 -1 v -1 2 b -2 n -2 -1 e Binomial v v 2 -1 2 e Gamma 2 n 1 -1 b z 1 2 -1 Gamma 2 n 1 b z 1 2 \$CellContext`v 2 -1 -1 2 s 0 v -1 2 -1 Binomial v s e 2 g s -1 g v Gamma 2 n 1 -1 b -1 2 c s c v z 1 2 -1 b -1 2 c s c v -2 n 1 e g v -1 2 s -1 b 2 c s -1 c v -2 n 1 Gamma 2 n 1 -1 b 2 c s -1 c v z 1 2 -1 -1 e -1 2 g s g v b 2 c s -1 c v -2 n 1 Gamma 2 n 1 b 2 c s -1 c v z 1 2 -1 -1 e 2 g s -1 g v b c v -1 2 s -2 n 1 Gamma 2 n 1 b c v -1 2 s z 1 2 n v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", "e_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", "g_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "b"]], " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["b", " ", SqrtBox["z"]]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", 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[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "-", RowBox[List["2", " ", "g", " ", "s"]], "+", RowBox[List["g", " ", "v"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", RowBox[List["2", " ", "g", " ", "s"]], "-", RowBox[List["g", " ", "v"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "+", RowBox[List["2", " ", "g", " ", "s"]], "-", RowBox[List["g", " ", "v"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18