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 Cosh

 http://functions.wolfram.com/01.20.21.4059.01

 Input Form

 Integrate[Sinh[b z^2 + d z + e] Cosh[f z + g]^v, z] == (1/b) (2^(-2 - v) E^(-(d^2/(4 b)) - e) Sqrt[Pi] Binomial[v, v/2] (Sqrt[-b] E^(d^2/(2 b)) Erfi[(Sqrt[-b] (d + 2 b z))/(2 b)] + Sqrt[b] E^(2 e) Erfi[(d + 2 b z)/(2 Sqrt[b])]) (1 - Mod[v, 2])) + 2^(-2 - v) Sqrt[Pi] Sum[Binomial[v, s] ((Sqrt[-b] E^(-e + g (2 s - v) + (d - 2 f s + f v)^2/(4 b)) Erfi[(-d + f (2 s - v) - 2 b z)/(2 Sqrt[-b])])/b + (E^(e + g (-2 s + v) - (d - 2 f s + f v)^2/(4 b)) Erfi[(d - f (2 s - v) + 2 b z)/(2 Sqrt[b])])/Sqrt[b] + (1/b) (E^(-e + g (-2 s + v) + (-d + f (-2 s + v))^2/(4 b)) (Sqrt[-b] Erfi[(-d - f (2 s - v) - 2 b z)/(2 Sqrt[-b])] + Sqrt[b] E^(2 (e + g (2 s - v) - (-d - 2 f s + f v)^2/(4 b))) Erfi[(d + f (2 s - v) + 2 b z)/(2 Sqrt[b])]))), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "b"], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["d", "2"], RowBox[List["4", " ", "b"]]]]], "-", "e"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "b"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["d", "2"], RowBox[List["2", " ", "b"]]]], " ", RowBox[List["Erfi", 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RowBox[List["2", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh ( b z 2 + d z + e ) cosh v ( g + f z ) z 1 b ( 2 - v - 2 - d 2 4 b - e π ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b 2 e erfi ( d + 2 b z 2 b ) + - b d 2 2 b erfi ( - b ( d + 2 b z ) 2 b ) ) ( 1 - v mod 2 \$CellContext`v 2 ) ) + 2 - v - 2 π s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - b ( d - 2 f s + f v ) 2 4 b - e + g ( 2 s - v ) erfi ( - d + f ( 2 s - v ) - 2 b z 2 - b ) b + 1 b ( ( f ( v - 2 s ) - d ) 2 4 b - e + g ( v - 2 s ) ( - b erfi ( - d - f ( 2 s - v ) - 2 b z 2 - b ) + b 2 ( - ( - d - 2 f s + f v ) 2 4 b + e + g ( 2 s - v ) ) erfi ( d + f ( 2 s - v ) + 2 b z 2 b ) ) ) + - ( d - 2 f s + f v ) 2 4 b + e + g ( v - 2 s ) erfi ( d - f ( 2 s - v ) + 2 b z 2 b ) b ) /; v + Condition z b z 2 d z e g f z v 1 b -1 2 -1 v -2 -1 d 2 4 b -1 -1 e 1 2 Binomial v v 2 -1 b 1 2 2 e Erfi d 2 b z 2 b 1 2 -1 -1 b 1 2 d 2 2 b -1 Erfi -1 b 1 2 d 2 b z 2 b -1 1 -1 \$CellContext`v 2 2 -1 v -2 1 2 s 0 v -1 2 -1 Binomial v s -1 b 1 2 d -1 2 f s f v 2 4 b -1 -1 e g 2 s -1 v Erfi -1 d f 2 s -1 v -1 2 b z 2 -1 b 1 2 -1 b -1 1 b -1 f v -1 2 s -1 d 2 4 b -1 -1 e g v -1 2 s -1 b 1 2 Erfi -1 d -1 f 2 s -1 v -1 2 b z 2 -1 b 1 2 -1 b 1 2 2 -1 -1 d -1 2 f s f v 2 4 b -1 e g 2 s -1 v Erfi d f 2 s -1 v 2 b z 2 b 1 2 -1 -1 d -1 2 f s f v 2 4 b -1 e g v -1 2 s Erfi d -1 f 2 s -1 v 2 b z 2 b 1 2 -1 b 1 2 -1 v SuperPlus [/itex]

 Rule Form

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

 2002-12-18