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 Cosh

 http://functions.wolfram.com/01.20.21.0777.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", " ", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "a"], "+", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "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", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]]]], RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["d_", "+", RowBox[List["c_", " ", "z_"]]]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "a"], "+", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "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", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]]]], RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]]], "+", FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]]]], RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]]], "+", FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]]]], RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]]], "+", FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", "m"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]], ")"]], " ", "z"]], "]"]]]], RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "j"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18