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 Cosh

 http://functions.wolfram.com/01.20.21.4084.01

 Input Form

 Integrate[Sinh[b z^2 + d z] Cosh[c z^2 + g]^v, z] == (1/b) ((2^(-2 - v) Sqrt[Pi] Binomial[v, v/2] (Sqrt[-b] E^(d^2/(2 b)) Erfi[(Sqrt[-b] (d + 2 b z))/(2 b)] + Sqrt[b] Erfi[(d + 2 b z)/(2 Sqrt[b])]) (1 - Mod[v, 2]))/ E^(d^2/(4 b))) + 2^(-2 - v) Sqrt[Pi] Sum[Binomial[v, s] ((E^(g (-2 s + v) + d^2/(-4 b + 8 c s - 4 c v)) Erfi[(d + 2 b z - 2 c (2 s - v) z)/(2 Sqrt[b + c (-2 s + v)])])/ Sqrt[b + c (-2 s + v)] + (E^(g (2 s - v) - d^2/(-4 b + 8 c s - 4 c v)) Sqrt[-b + 2 c s - c v] Erfi[(-d - 2 b z + 2 c (2 s - v) z)/(2 Sqrt[-b + 2 c s - c v])])/ (b + c (-2 s + v)) + (1/(b + 2 c s - c v)) (E^(g (-2 s + v) - d^2/(-4 b - 8 c s + 4 c v)) (Sqrt[-b - 2 c s + c v] Erfi[(-d - 2 b z - 2 c (2 s - v) z)/ (2 Sqrt[-b - 2 c s + c v])] + E^(2 (g (2 s - v) + d^2/(-4 b - 8 c s + 4 c v))) Sqrt[b + 2 c s - c v] Erfi[(d + 2 b z + 2 c (2 s - v) z)/ (2 Sqrt[b + 2 c s - c 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[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", "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["-", FractionBox[SuperscriptBox["d", "2"], RowBox[List["4", " ", "b"]]]]]], " ", 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[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", FractionBox[SuperscriptBox["d", "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], RowBox[List["2", 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 MathML Form

 sinh ( b z 2 + d z ) cosh v ( c z 2 + g ) z 2 - v - 2 - d 2 4 b 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 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 ) b + 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]]]]] ( g ( 2 s - v ) - d 2 - 4 b + 8 c s - 4 c v - b + 2 c s - c v erfi ( - d - 2 b z + 2 c ( 2 s - v ) z 2 - b + 2 c s - c v ) b + c ( v - 2 s ) + 1 b + 2 c s - c v ( g ( v - 2 s ) - d 2 - 4 b - 8 c s + 4 c v ( - b - 2 c s + c v erfi ( - d - 2 b z - 2 c ( 2 s - v ) z 2 - b - 2 c s + c v ) + 2 ( d 2 - 4 b - 8 c s + 4 c v + g ( 2 s - v ) ) b + 2 c s - c v erfi ( d + 2 b z + 2 c ( 2 s - v ) z 2 b + 2 c s - c v ) ) ) + d 2 - 4 b + 8 c s - 4 c v + g ( v - 2 s ) erfi ( d + 2 b z - 2 c ( 2 s - v ) z 2 b + c ( v - 2 s ) ) b + c ( v - 2 s ) ) /; v + Condition z b z 2 d z c z 2 g v 2 -1 v -2 -1 d 2 4 b -1 1 2 Binomial v v 2 -1 b 1 2 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 b -1 2 -1 v -2 1 2 s 0 v -1 2 -1 Binomial v s g 2 s -1 v -1 d 2 -4 b 8 c s -1 4 c v -1 -1 b 2 c s -1 c v 1 2 Erfi -1 d -1 2 b z 2 c 2 s -1 v z 2 -1 b 2 c s -1 c v 1 2 -1 b c v -1 2 s -1 1 b 2 c s -1 c v -1 g v -1 2 s -1 d 2 -4 b -1 8 c s 4 c v -1 -1 b -1 2 c s c v 1 2 Erfi -1 d -1 2 b z -1 2 c 2 s -1 v z 2 -1 b -1 2 c s c v 1 2 -1 2 d 2 -4 b -1 8 c s 4 c v -1 g 2 s -1 v b 2 c s -1 c v 1 2 Erfi d 2 b z 2 c 2 s -1 v z 2 b 2 c s -1 c v 1 2 -1 d 2 -4 b 8 c s -1 4 c v -1 g v -1 2 s Erfi d 2 b z -1 2 c 2 s -1 v z 2 b c v -1 2 s 1 2 -1 b c v -1 2 s 1 2 -1 v SuperPlus [/itex]

 Rule Form

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

 2002-12-18