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 Cosh

 http://functions.wolfram.com/01.20.21.1148.01

 Input Form

 Integrate[z^n Cos[b Sqrt[z]]^m Cosh[c Sqrt[z] + g], z] == ((-2^(-m)) Binomial[m, m/2] (E^g Gamma[2 (1 + n), (-c) Sqrt[z]] + Gamma[2 (1 + n), c Sqrt[z]]/E^g) (1 - Mod[m, 2]))/(-c)^(2 (1 + n)) - Sum[Binomial[m, k] ((E^g Gamma[2 (1 + n), (-c + 2 I b k - I b m) Sqrt[z]])/ (-c + 2 I b k - I b m)^(2 (1 + n)) + Gamma[2 (1 + n), (c + 2 I b k - I b m) Sqrt[z]]/ (E^g (c + 2 I b k - I b m)^(2 (1 + n))) + (E^g Gamma[2 (1 + n), (-c - 2 I b k + I b m) Sqrt[z]])/ (-c - 2 I b k + I b m)^(2 (1 + n)) + Gamma[2 (1 + n), (c - 2 I b k + I b m) Sqrt[z]]/ (E^g (c - 2 I b k + I b m)^(2 (1 + n)))), {k, 0, Floor[(1/2) (-1 + m)]}]/2^m /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "m"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "c"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "g"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "c"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", 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[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "g"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", "g"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 z n cos m ( b z ) cosh ( z c + g ) z - 2 - m ( 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]]]]] ( g Γ ( 2 ( n + 1 ) , - c z ) + - g Γ ( 2 ( n + 1 ) , c z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) ( - c ) - 2 ( n + 1 ) - 2 - m k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( g Γ ( 2 ( n + 1 ) , ( - c - 2 b k + b m ) z ) ( - c - 2 b k + b m ) - 2 ( n + 1 ) + - g ( c - 2 b k + b m ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( c - 2 b k + b m ) z ) + g ( - c + 2 b k - b m ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - c + 2 b k - b m ) z ) + - g ( c + 2 b k - b m ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( c + 2 b k - b m ) z ) ) /; n m + Condition z z n b z 1 2 m z 1 2 c g -1 2 -1 m Binomial m m 2 -1 g Gamma 2 n 1 -1 c z 1 2 -1 g Gamma 2 n 1 c z 1 2 1 -1 \$CellContext`m 2 -1 c -2 n 1 -1 2 -1 m k 0 m -1 2 -1 Binomial m k g Gamma 2 n 1 -1 c -1 2 b k b m z 1 2 -1 c -1 2 b k b m -2 n 1 -1 g c -1 2 b k b m -2 n 1 Gamma 2 n 1 c -1 2 b k b m z 1 2 g -1 c 2 b k -1 b m -2 n 1 Gamma 2 n 1 -1 c 2 b k -1 b m z 1 2 -1 g c 2 b k -1 b m -2 n 1 Gamma 2 n 1 c 2 b k -1 b m z 1 2 n m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", "m"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "c"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "g"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "c"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", 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[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "g"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", "g"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18