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 Cosh

 http://functions.wolfram.com/01.20.21.0984.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z]]^m Cosh[c Sqrt[z]], z] == ((-(-(1/2))^m) Binomial[m, m/2] (Gamma[2 (1 + n), (-c) Sqrt[z]] + Gamma[2 (1 + n), c Sqrt[z]]) (1 - Mod[m, 2]))/c^(2 (1 + n)) - Sum[(-1)^k Binomial[m, k] (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]]/(c + 2 I b k - I b m)^ (2 (1 + n)) + ((-1)^m 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)) + ((-1)^m 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))), {k, 0, Floor[(1/2) (-1 + m)]}]/ (2 I)^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["Sin", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "m"]]], " ", SuperscriptBox["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["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "c"]], " ", SqrtBox["z"]]]]], "]"]], "+", 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[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]"]], ")"]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[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[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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", 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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", 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 sin m ( b z ) cosh ( c z ) z - k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - 1 ) m Γ ( 2 ( n + 1 ) , ( - c - 2 b k + b m ) z ) ( - c - 2 b k + b m ) - 2 ( n + 1 ) + ( - 1 ) m ( c - 2 b k + b m ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( c - 2 b k + b m ) z ) + ( - c + 2 b k - b m ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - c + 2 b k - b m ) z ) + ( c + 2 b k - b m ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( c + 2 b k - b m ) z ) ) ( 2 ) - m - ( - 1 2 ) m c - 2 ( n + 1 ) ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( Γ ( 2 ( n + 1 ) , - c z ) + Γ ( 2 ( n + 1 ) , c z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) /; n m + Condition z z n b z 1 2 m c z 1 2 -1 k 0 m -1 2 -1 -1 k Binomial m k -1 m 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 m c -1 2 b k b m -2 n 1 Gamma 2 n 1 c -1 2 b k b m z 1 2 -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 c 2 b k -1 b m -2 n 1 Gamma 2 n 1 c 2 b k -1 b m z 1 2 2 -1 m -1 -1 1 2 m c -2 n 1 Binomial m m 2 -1 Gamma 2 n 1 -1 c z 1 2 Gamma 2 n 1 c z 1 2 1 -1 \$CellContext`m 2 n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18