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 Cosh

 http://functions.wolfram.com/01.20.21.0635.01

 Input Form

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

 Standard Form

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

 MathML Form

 sin m ( z b + e ) cosh ( c z ) z 2 1 - m 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 ( - c + b ( 2 k - m ) ) 2 ( cos ( z ( - c + b ( 2 k - m ) ) - e ( m - 2 k ) + m π 2 ) + ( - c + b ( 2 k - m ) ) z sin ( z ( - c + b ( 2 k - m ) ) - e ( m - 2 k ) + m π 2 ) ) + 1 ( c + b ( 2 k - m ) ) 2 ( cos ( - z ( c + b ( 2 k - m ) ) + e ( m - 2 k ) - m π 2 ) - ( c + b ( 2 k - m ) ) z sin ( - z ( c + b ( 2 k - m ) ) + e ( m - 2 k ) - m π 2 ) ) ) - 2 1 - 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( cosh ( c z ) - c z sinh ( c z ) ) c 2 /; m + Condition z z 1 2 b e m c z 1 2 2 1 -1 m k 0 m -1 2 -1 -1 k Binomial m k 1 -1 c b 2 k -1 m 2 -1 z 1 2 -1 c b 2 k -1 m -1 e m -1 2 k m 2 -1 -1 c b 2 k -1 m z 1 2 z 1 2 -1 c b 2 k -1 m -1 e m -1 2 k m 2 -1 1 c b 2 k -1 m 2 -1 -1 z 1 2 c b 2 k -1 m e m -1 2 k -1 m 2 -1 -1 c b 2 k -1 m z 1 2 -1 z 1 2 c b 2 k -1 m e m -1 2 k -1 m 2 -1 -1 2 1 -1 m Binomial m m 2 -1 1 -1 \$CellContext`m 2 c z 1 2 -1 c z 1 2 c z 1 2 c 2 -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18