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 Cosh

 http://functions.wolfram.com/01.20.21.4091.01

 Input Form

 Integrate[Sinh[d z + e] Cosh[c Sqrt[z] + f z]^v, z] == (Cosh[e + d z] Binomial[v, v/2] (1 - Mod[v, 2]))/(2^v d) + 2^(-2 - v) Sum[Binomial[v, s] (E^e ((4 Cosh[e + c (2 s - v) Sqrt[z] + (d + 2 f s - f v) z])/ (E^e (d + 2 f s - f v)) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (-d + f (-2 s + v)) Sqrt[z])/ (2 Sqrt[d - f (-2 s + v)])])/E^((c^2 (-2 s + v)^2)/ (4 (d - f (-2 s + v))))/(d - f (-2 s + v))^(3/2) + (c E^(-2 e - (c^2 (-2 s + v)^2)/(4 (-d + f (-2 s + v)))) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (-d + f (-2 s + v)) Sqrt[z])/ (2 Sqrt[-d + f (-2 s + v)])])/(-d + f (-2 s + v))^(3/2)) - (-((4 E^e Cosh[e - c (2 s - v) Sqrt[z] + (d - 2 f s + f v) z])/ (d + f (-2 s + v))) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z])/ (2 Sqrt[-d - f (-2 s + v)])])/E^((c^2 (-2 s + v)^2)/ (4 (-d - f (-2 s + v))))/(-d - f (-2 s + v))^(3/2) + (c E^(2 e - (c^2 (-2 s + v)^2)/(4 (d + f (-2 s + v)))) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (d + f (-2 s + v)) Sqrt[z])/ (2 Sqrt[d + f (-2 s + v)])])/(d + f (-2 s + v))^(3/2))/E^e), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

 Standard Form

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SqrtBox[RowBox[List["d", "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh ( e + d z ) cosh v ( z c + f z ) z 2 - v ( 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]]]]] cosh ( e + d z ) ( 1 - v mod 2 \$CellContext`v 2 ) d + 2 - v - 2 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]]]]] ( e ( 4 - e cosh ( e + ( d + 2 f s - f v ) z + c ( 2 s - v ) z ) d + 2 f s - f v + c - c 2 ( v - 2 s ) 2 4 ( f ( v - 2 s ) - d ) - 2 e π ( v - 2 s ) erfi ( c ( v - 2 s ) + 2 ( f ( v - 2 s ) - d ) z 2 f ( v - 2 s ) - d ) ( f ( v - 2 s ) - d ) 3 / 2 - c - c 2 ( v - 2 s ) 2 4 ( d - f ( v - 2 s ) ) π ( v - 2 s ) erfi ( c ( v - 2 s ) + 2 ( f ( v - 2 s ) - d ) z 2 d - f ( v - 2 s ) ) ( d - f ( v - 2 s ) ) 3 / 2 ) - - e ( - 4 e cosh ( e + ( d - 2 f s + f v ) z - c ( 2 s - v ) z ) d + f ( v - 2 s ) + c 2 e - c 2 ( v - 2 s ) 2 4 ( d + f ( v - 2 s ) ) π ( v - 2 s ) erfi ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z 2 d + f ( v - 2 s ) ) ( d + f ( v - 2 s ) ) 3 / 2 - c - c 2 ( v - 2 s ) 2 4 ( - d - f ( v - 2 s ) ) π ( v - 2 s ) erfi ( c ( v - 2 s ) + 2 ( d + f ( v - 2 s ) ) z 2 - d - f ( v - 2 s ) ) ( - d - f ( v - 2 s ) ) 3 / 2 ) ) /; v + Condition z e d z z 1 2 c f z v 2 -1 v Binomial v v 2 -1 e d z 1 -1 \$CellContext`v 2 d -1 2 -1 v -2 s 0 v -1 2 -1 Binomial v s e 4 -1 e e d 2 f s -1 f v z c 2 s -1 v z 1 2 d 2 f s -1 f v -1 c -1 c 2 v -1 2 s 2 4 f v -1 2 s -1 d -1 -1 2 e 1 2 v -1 2 s Erfi c v -1 2 s 2 f v -1 2 s -1 d z 1 2 2 f v -1 2 s -1 d 1 2 -1 f v -1 2 s -1 d 3 2 -1 -1 c -1 c 2 v -1 2 s 2 4 d -1 f v -1 2 s -1 1 2 v -1 2 s Erfi c v -1 2 s 2 f v -1 2 s -1 d z 1 2 2 d -1 f v -1 2 s 1 2 -1 d -1 f v -1 2 s 3 2 -1 -1 -1 e -1 4 e e d -1 2 f s f v z -1 c 2 s -1 v z 1 2 d f v -1 2 s -1 c 2 e -1 c 2 v -1 2 s 2 4 d f v -1 2 s -1 1 2 v -1 2 s Erfi c v -1 2 s 2 d f v -1 2 s z 1 2 2 d f v -1 2 s 1 2 -1 d f v -1 2 s 3 2 -1 -1 c -1 c 2 v -1 2 s 2 4 -1 d -1 f v -1 2 s -1 1 2 v -1 2 s Erfi c v -1 2 s 2 d f v -1 2 s z 1 2 2 -1 d -1 f v -1 2 s 1 2 -1 -1 d -1 f v -1 2 s 3 2 -1 v SuperPlus [/itex]

 Rule Form

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

 2002-12-18