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 Cosh

 http://functions.wolfram.com/01.20.21.3651.01

 Input Form

 Integrate[z^n Cos[c z + d]^m Cosh[a z]^\[Nu], z] == (Binomial[m, m/2] n! (1 - Mod[m, 2]) Cosh[a z]^\[Nu] Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (a \[Nu])^(-1 - j) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, 1 + j], -\[Nu]}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, 1 + j]}, -E^(-2 a z)]), {j, 0, n}])/ (2^m (1 + E^(-2 a z))^\[Nu]) + (n! Cosh[a z]^\[Nu] Sum[Binomial[m, k] (E^((1/2) I (4 d k - 2 d m + 4 c k z - 2 c m z)) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) ((-I) c (-2 k + m) + a \[Nu])^ (-1 - j) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], -\[Nu]}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, -E^(-2 a z)]), {j, 0, n}] + Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (I c (-2 k + m) + a \[Nu])^(-1 - j) HypergeometricPFQ[ {Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], -\[Nu]}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, -E^(-2 a z)]), {j, 0, n}]/E^((1/2) I (4 d k - 2 d m + 4 c k z - 2 c m z))), {k, 0, Floor[(1/2) (-1 + m)]}])/ (2^m (1 + E^(-2 a z))^\[Nu]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == -(\[Nu]/2) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (I (c (-2 k + m) + I a \[Nu]))/(2 a) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (I (2 c k - c m + I a \[Nu]))/(2 a) && 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[RowBox[List["c", " ", "z"]], "+", "d"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", 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RowBox[List["2", " ", "a"]]]]], "\[And]", 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 ( d + c z ) cosh ν ( a z ) 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]]]]] cosh ν ( a z ) n ! ( 1 - m mod 2 \$CellContext`m 2 ) ( j = 0 n ( - 1 ) j z n - j ( a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - ν 2 , , - ν 2 , - ν ; 1 - ν 2 , , 1 - ν 2 ; - - 2 a z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ( 1 + - 2 a z ) - ν + 2 - m cosh ν ( a z ) n ! ( 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]]]]] ( - 1 2 ( 4 d k + 4 c z k - 2 d m - 2 c m z ) j = 0 n ( - 1 ) j z n - j ( c ( m - 2 k ) + a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( ( 2 c k - c m + a ν ) 2 a , , ( 2 c k - c m + a ν ) 2 a , - ν ; ( 2 c k - c m + a ν ) 2 a + 1 , , ( 2 c k - c m + a ν ) 2 a + 1 ; - - 2 a z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "k"]], "-", RowBox[List["c", " ", "m"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "k"]], "-", RowBox[List["c", " ", "m"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "k"]], "-", RowBox[List["c", " ", "m"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "k"]], "-", RowBox[List["c", " ", "m"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 1 2 ( 4 d k + 4 c z k - 2 d m - 2 c m z ) j = 0 n ( - 1 ) j z n - j ( a ν - c ( m - 2 k ) ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( ( c ( m - 2 k ) + a ν ) 2 a , , ( c ( m - 2 k ) + a ν ) 2 a , - ν ; ( c ( m - 2 k ) + a ν ) 2 a + 1 , , ( c ( m - 2 k ) + a ν ) 2 a + 1 ; - - 2 a z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ) ( 1 + - 2 a z ) - ν /; n m + Condition z z n d c z m a z ν 2 -1 m Binomial m m 2 -1 a z ν n 1 -1 \$CellContext`m 2 j 0 n -1 j z n -1 j a ν -1 j -1 n -1 j -1 HypergeometricPFQ -1 ν 2 -1 -1 ν 2 -1 -1 ν 1 -1 ν 2 -1 1 -1 ν 2 -1 -1 -2 a z 1 -2 a z -1 ν 2 -1 m a z ν n k 0 m -1 2 -1 Binomial m k -1 1 2 4 d k 4 c z k -1 2 d m -1 2 c m z j 0 n -1 j z n -1 j c m -1 2 k a ν -1 j -1 n -1 j -1 HypergeometricPFQ 2 c k -1 c m a ν 2 a -1 2 c k -1 c m a ν 2 a -1 -1 ν 2 c k -1 c m a ν 2 a -1 1 2 c k -1 c m a ν 2 a -1 1 -1 -2 a z 1 2 4 d k 4 c z k -1 2 d m -1 2 c m z j 0 n -1 j z n -1 j a ν -1 c m -1 2 k -1 j -1 n -1 j -1 HypergeometricPFQ c m -1 2 k a ν 2 a -1 c m -1 2 k a ν 2 a -1 -1 ν c m -1 2 k a ν 2 a -1 1 c m -1 2 k a ν 2 a -1 1 -1 -2 a z 1 -2 a z -1 ν n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18