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 Cosh

 http://functions.wolfram.com/01.20.21.0262.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2002-12-18