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 Cosh

 http://functions.wolfram.com/01.20.21.1570.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["d", " ", "z"]], "+", "e"]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "c"], RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "c"]]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["-", "c"]]], " ", "z"]], "]"]]]], "-", RowBox[List[SqrtBox["c"], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox["c"], " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SqrtBox[RowBox[List["-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["e", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], "c"]]], "-", RowBox[List[FractionBox["1", "c"], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "e"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "m"]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["e", " ", "m"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]], ")"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "c"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SqrtBox["c"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "e", " ", "k"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]]]], ")"]]]], ")"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["e", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["c"]]]], "]"]]]], SqrtBox["c"]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18