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 Cosh

 http://functions.wolfram.com/01.20.21.0473.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2002-12-18