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 Cosh

 http://functions.wolfram.com/01.20.21.4945.01

 Input Form

 Integrate[Sinh[d z] Cosh[c Sqrt[z] + g]^v, z] == (Cosh[d z] Binomial[v, v/2] (1 - Mod[v, 2]))/(2^v d) + 2^(-2 - v) Sum[Binomial[v, s] (((2 E^(c (2 s - v) Sqrt[z] - d z) (E^(2 (-2 s + v) (g + c Sqrt[z])) + E^(2 d z)))/d + (1/(-d)^(3/2)) (c E^(2 g (-2 s + v) + (c^2 (-2 s + v)^2)/(4 d)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) - 2 d Sqrt[z])/(2 Sqrt[-d])]) - (1/d^(3/2)) ((c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) - 2 d Sqrt[z])/(2 Sqrt[d])])/E^((c^2 (-2 s + v)^2)/(4 d))))/ E^(g (-2 s + v)) - (-((4 E^(g (-2 s + v)) Cosh[g (-2 s + v) + c (-2 s + v) Sqrt[z] + d z])/d) - (c E^((c^2 (-2 s + v)^2)/(4 d)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 d Sqrt[z])/(2 Sqrt[-d])])/(-d)^(3/2) + (1/d^(3/2)) (c E^(2 g (-2 s + v) - (c^2 (-2 s + v)^2)/(4 d)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 d Sqrt[z])/ (2 Sqrt[d])]))/E^(g (-2 s + v))), {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["d", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", "g"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "v"]]], " ", RowBox[List["Cosh", "[", RowBox[List["d", " ", "z"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "d"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], 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"s"]], "+", "v"]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["d_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", "g_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "v"]]], " ", RowBox[List["Cosh", "[", RowBox[List["d", " ", "z"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "d"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", 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 Date Added to functions.wolfram.com (modification date)

 2003-08-21