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 Cosh

 http://functions.wolfram.com/01.20.21.0476.01

 Input Form

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

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

 MathML Form

 z b + e + d z cosh ( z c + f z ) z - ( b - c ) - b 2 - 2 c b + c 2 - 4 e ( d - f ) 4 ( d - f ) π erfi ( b - c + 2 ( d - f ) z 2 d - f ) 4 ( d - f ) 3 / 2 + z ( b - c ) + e + ( d - f ) z 2 ( d - f ) + z ( b + c ) + e + ( d + f ) z 2 ( d + f ) - ( b + c ) - b 2 + 2 c b + c 2 - 4 e ( d + f ) 4 ( d + f ) π erfi ( b + c + 2 ( d + f ) z 2 d + f ) 4 ( d + f ) 3 / 2 z z 1 2 b e d z z 1 2 c f z -1 b -1 c -1 b 2 -1 2 c b c 2 -1 4 e d -1 f 4 d -1 f -1 1 2 Erfi b -1 c 2 d -1 f z 1 2 2 d -1 f 1 2 -1 4 d -1 f 3 2 -1 z 1 2 b -1 c e d -1 f z 2 d -1 f -1 z 1 2 b c e d f z 2 d f -1 -1 b c -1 b 2 2 c b c 2 -1 4 e d f 4 d f -1 1 2 Erfi b c 2 d f z 1 2 2 d f 1 2 -1 4 d f 3 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_"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]], " ", "z"]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]]]]], "+", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], " ", "z"]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["2", " ", "b", " ", "c"]], "+", SuperscriptBox["c", "2"], "-", RowBox[List["4", " ", "e", " ", RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]]]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "-", "c", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["d", "-", "f"]]]]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]], RowBox[List["3", "/", "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["2", " ", "b", " ", "c"]], "+", SuperscriptBox["c", "2"], "-", RowBox[List["4", " ", "e", " ", RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]]]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["d", "+", "f"]]]]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], RowBox[List["3", "/", "2"]]]]]]]]]]]]

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

 2002-12-18