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 Cosh

 http://functions.wolfram.com/01.20.21.0267.01

 Input Form

 Integrate[E^(b Sqrt[z] + d z + e) Cosh[c Sqrt[z] + g], z] == (1/(4 d^(3/2))) ((2 Sqrt[d] E^((b^2 + (c - 2 d Sqrt[z])^2 + 2 b (c + 2 d Sqrt[z]))/(4 d)) (1 + E^(2 (g + c Sqrt[z]))) - (b - c) E^((b c)/d) Sqrt[Pi] Erfi[(b - c + 2 d Sqrt[z])/(2 Sqrt[d])] - (b + c) E^(2 g) Sqrt[Pi] Erfi[(b + c + 2 d Sqrt[z])/(2 Sqrt[d])])/ E^((b^2 + 2 b c + c^2 - 4 d e + 4 d g)/(4 d)))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["d", RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", "b", " ", "c"]], "+", SuperscriptBox["c", "2"], "-", RowBox[List["4", " ", "d", " ", "e"]], "+", RowBox[List["4", " ", "d", " ", "g"]]]], RowBox[List["4", " ", "d"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["d"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], ")"]], "2"], "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], ")"]]]]]], RowBox[List["4", " ", "d"]]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], ")"]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["b", " ", "c"]], "d"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "-", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "g"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z b + e + d z cosh ( z c + g ) z 1 4 d 3 / 2 ( - b 2 + 2 c b + c 2 - 4 d e + 4 d g 4 d ( 2 d b 2 + 2 ( c + 2 d z ) b + ( c - 2 d z ) 2 4 d ( 1 + 2 ( z c + g ) ) - ( b - c ) b c d π erfi ( b - c + 2 d z 2 d ) - ( b + c ) 2 g π erfi ( b + c + 2 d z 2 d ) ) ) z z 1 2 b e d z z 1 2 c g 1 4 d 3 2 -1 -1 b 2 2 c b c 2 -1 4 d e 4 d g 4 d -1 2 d 1 2 b 2 2 c 2 d z 1 2 b c -1 2 d z 1 2 2 4 d -1 1 2 z 1 2 c g -1 b -1 c b c d -1 1 2 Erfi b -1 c 2 d z 1 2 2 d 1 2 -1 -1 b c 2 g 1 2 Erfi b c 2 d z 1 2 2 d 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", "b", " ", "c"]], "+", SuperscriptBox["c", "2"], "-", RowBox[List["4", " ", "d", " ", "e"]], "+", RowBox[List["4", " ", "d", " ", "g"]]]], RowBox[List["4", " ", "d"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["d"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], ")"]], "2"], "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], ")"]]]]]], RowBox[List["4", " ", "d"]]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], ")"]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["b", " ", "c"]], "d"]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "-", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "g"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["d", RowBox[List["3", "/", "2"]]]]]]]]]]

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

 2002-12-18