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 Cosh

 http://functions.wolfram.com/01.20.21.3967.01

 Input Form

 Integrate[z^n E^(p z) Cos[c z + d] Cosh[a z]^\[Nu], z] == ((n!/2) Cosh[a z]^\[Nu] (E^(I d + (p + I c) z) Sum[(((-1)^j z^(n - j))/((n - j)! (p + I c - a \[Nu])^(j + 1))) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, j + 1], -\[Nu]}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, j + 1]}, -E^(2 a z)], {j, 0, n}] + E^((-I) d + (p - I c) z) Sum[(((-1)^j z^(n - j))/ ((n - j)! (p - I c - a \[Nu])^(j + 1))) HypergeometricPFQ[ {Subscript[d, 1], \[Ellipsis], Subscript[d, j + 1], -\[Nu]}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, j + 1]}, -E^(2 a z)], {j, 0, n}]))/(1 + E^(2 a z))^\[Nu] /; Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (p + I c - a \[Nu])/(2 a) && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == (p - I c - a \[Nu])/(2 a) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z cos ( d + c z ) cosh ν ( a z ) z 1 2 n ! cosh ν ( a z ) ( 1 + 2 a z ) - ν ( - d + ( p - c ) z j = 0 n ( ( - 1 ) j z n - j ) ( n - j ) ! ( p - c - a ν ) j + 1 j + 2 F j + 1 ( - c + p - a ν 2 a , , - c + p - a ν 2 a , - ν ; - c + p - a ν 2 a + 1 , , - c + p - 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[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "-", RowBox[List["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[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + d + ( p + c ) z j = 0 n ( ( - 1 ) j z n - j ) ( n - j ) ! ( p + c - a ν ) j + 1 j + 2 F j + 1 ( c + p - a ν 2 a , , c + p - a ν 2 a , - ν ; c + p - a ν 2 a + 1 , , c + p - 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[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "-", RowBox[List["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[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "a", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n Condition z z n p z d c z a z ν 1 2 n a z ν 1 2 a z -1 ν -1 d p -1 c z j 0 n -1 j z n -1 j n -1 j p -1 c -1 a ν j 1 -1 HypergeometricPFQ -1 c p -1 a ν 2 a -1 -1 c p -1 a ν 2 a -1 -1 ν -1 c p -1 a ν 2 a -1 1 -1 c p -1 a ν 2 a -1 1 -1 2 a z d p c z j 0 n -1 j z n -1 j n -1 j p c -1 a ν j 1 -1 HypergeometricPFQ c p -1 a ν 2 a -1 c p -1 a ν 2 a -1 -1 ν c p -1 a ν 2 a -1 1 c p -1 a ν 2 a -1 1 -1 2 a z n [/itex]

 Rule Form

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

 2002-12-18