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 Cosh

 http://functions.wolfram.com/01.20.21.0264.01

 Input Form

 Integrate[a^(b z^2 + d z + e) Cosh[c z^2 + g], z] == (a^(e + (b d^2 Log[a]^2)/(2 c^2 - 2 b^2 Log[a]^2)) Sqrt[Pi] ((-E^((d^2 Log[a]^2)/(4 c + 4 b Log[a]))) Erfi[(-2 c z + (d + 2 b z) Log[a])/(2 Sqrt[-c + b Log[a]])] Sqrt[-c + b Log[a]] (c + b Log[a]) (Cosh[g] - Sinh[g]) + E^((d^2 Log[a]^2)/(-4 c + 4 b Log[a])) Erfi[(2 c z + (d + 2 b z) Log[a])/(2 Sqrt[c + b Log[a]])] (c - b Log[a]) Sqrt[c + b Log[a]] (Cosh[g] + Sinh[g])))/(4 (c^2 - b^2 Log[a]^2))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List["e", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List[RowBox[List["4", " ", "c"]], "+", RowBox[List["4", " ", "b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "-", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "c"]], "+", RowBox[List["4", " ", "b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "+", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "-", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 a b z 2 + d z + e cosh ( c z 2 + g ) z 1 4 ( c 2 - b 2 log 2 ( a ) ) ( a b d 2 log 2 ( a ) 2 c 2 - 2 b 2 log 2 ( a ) + e π ( d 2 log 2 ( a ) 4 b log ( a ) - 4 c erfi ( 2 c z + ( d + 2 b z ) log ( a ) 2 c + b log ( a ) ) ( c - b log ( a ) ) c + b log ( a ) ( cosh ( g ) + sinh ( g ) ) - d 2 log 2 ( a ) 4 c + 4 b log ( a ) erfi ( ( d + 2 b z ) log ( a ) - 2 c z 2 b log ( a ) - c ) b log ( a ) - c ( c + b log ( a ) ) ( cosh ( g ) - sinh ( g ) ) ) ) z a b z 2 d z e c z 2 g 1 4 c 2 -1 b 2 a 2 -1 a b d 2 a 2 2 c 2 -1 2 b 2 a 2 -1 e 1 2 d 2 a 2 4 b a -1 4 c -1 Erfi 2 c z d 2 b z a 2 c b a 1 2 -1 c -1 b a c b a 1 2 g g -1 d 2 a 2 4 c 4 b a -1 Erfi d 2 b z a -1 2 c z 2 b a -1 c 1 2 -1 b a -1 c 1 2 c b a g -1 g [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["a_", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List["e", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List[RowBox[List["2", " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List[RowBox[List["4", " ", "c"]], "+", RowBox[List["4", " ", "b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "-", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "c"]], "+", RowBox[List["4", " ", "b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "g", "]"]], "+", RowBox[List["Sinh", "[", "g", "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "-", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]]]], ")"]]]]]]]]]

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

 2002-12-18