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 Cosh

 http://functions.wolfram.com/01.20.21.0495.01

 Input Form

 Integrate[a^(b Sqrt[z] + d z) Cosh[c Sqrt[z] + f z + g], z] == -((Sqrt[Pi] Erfi[(-c + b Log[a] + 2 Sqrt[z] (-f + d Log[a]))/ (2 Sqrt[-f + d Log[a]])] (-c + b Log[a]))/ E^((c^2 - 2 b c Log[a] + b^2 Log[a]^2 + 4 (-f + d Log[a]) g)/ (4 (-f + d Log[a]))))/(4 (-f + d Log[a])^(3/2)) + (a^(b Sqrt[z] + d z) E^(-g - c Sqrt[z] - f z))/(2 (-f + d Log[a])) - (Sqrt[Pi] Erfi[(c + b Log[a] + 2 Sqrt[z] (f + d Log[a]))/ (2 Sqrt[f + d Log[a]])] (c + b Log[a]))/ E^((c^2 + 2 b c Log[a] + b^2 Log[a]^2 - 4 (f + d Log[a]) g)/ (4 (f + d Log[a])))/(4 (f + d Log[a])^(3/2)) + (a^(b Sqrt[z] + d z) E^(g + c Sqrt[z] + f z))/(2 (f + 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"]]], "+", RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", 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", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", "g"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "g"]], "-", RowBox[List["c", " ", SqrtBox["z"]]], "-", RowBox[List["f", " ", "z"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]], "-", RowBox[List[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", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", "g"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]]]]]]]

 MathML Form

 a z b + d z cosh ( z c + g + f z ) z - z c - g - f z a z b + d z 2 ( d log ( a ) - f ) + z c + g + f z a z b + d z 2 ( f + d log ( a ) ) - - c 2 - 2 b log ( a ) c + b 2 log 2 ( a ) + 4 ( d log ( a ) - f ) g 4 ( d log ( a ) - f ) π erfi ( - c + b log ( a ) + 2 z ( d log ( a ) - f ) 2 d log ( a ) - f ) ( b log ( a ) - c ) 4 ( d log ( a ) - f ) 3 / 2 - - c 2 + 2 b log ( a ) c + b 2 log 2 ( a ) - 4 ( f + d log ( a ) ) g 4 ( f + d log ( a ) ) π erfi ( c + b log ( a ) + 2 z ( f + d log ( a ) ) 2 f + d log ( a ) ) ( c + b log ( a ) ) 4 ( f + d log ( a ) ) 3 / 2 z a z 1 2 b d z z 1 2 c g f z -1 z 1 2 c -1 g -1 f z a z 1 2 b d z 2 d a -1 f -1 z 1 2 c g f z a z 1 2 b d z 2 f d a -1 -1 -1 c 2 -1 2 b a c b 2 a 2 4 d a -1 f g 4 d a -1 f -1 1 2 Erfi -1 c b a 2 z 1 2 d a -1 f 2 d a -1 f 1 2 -1 b a -1 c 4 d a -1 f 3 2 -1 -1 -1 c 2 2 b a c b 2 a 2 -1 4 f d a g 4 f d a -1 1 2 Erfi c b a 2 z 1 2 f d a 2 f d a 1 2 -1 c b a 4 f d a 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_"]]], "+", RowBox[List["d_", " ", "z_"]]]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", 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", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", "g"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "g"]], "-", RowBox[List["c", " ", SqrtBox["z"]]], "-", RowBox[List["f", " ", "z"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]], "-", 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", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], " ", "g"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]]]]]]]

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

 2002-12-18