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 Cosh

 http://functions.wolfram.com/01.20.21.0491.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18