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 Cosh

 http://functions.wolfram.com/01.20.21.3764.01

 Input Form

 Integrate[E^(p z) Sin[c z]^\[Mu] Cosh[b + a z]^v, z] == (1/(p - I c \[Mu])) ((E^(p z) Binomial[v, v/2] Hypergeometric2F1[ -((I (p - I c \[Mu]))/(2 c)), -\[Mu], (1/2) (2 - (I p)/c - \[Mu]), E^(2 I c z)] (1 - Mod[v, 2]) Sin[c z]^\[Mu])/ (2^v (1 - E^(2 I c z))^\[Mu])) + (Sin[c z]^\[Mu] Sum[E^(b (-2 k + v)) Binomial[v, k] ((E^(-2 b (-2 k + v) + (p - a (-2 k + v)) z) Hypergeometric2F1[ -((I (p - a (-2 k + v) - I c \[Mu]))/(2 c)), -\[Mu], (1/2) (2 - (I (p - a (-2 k + v)))/c - \[Mu]), E^(2 I c z)])/ (p - a (-2 k + v) - I c \[Mu]) + (E^((p + a (-2 k + v)) z) Hypergeometric2F1[-((I (p + a (-2 k + v) - I c \[Mu]))/(2 c)), -\[Mu], (1/2) (2 - (I (p + a (-2 k + v)))/c - \[Mu]), E^(2 I c z)])/(p + a (-2 k + v) - I c \[Mu])), {k, 0, Floor[(1/2) (-1 + v)]}])/(2^v (1 - E^(2 I c z))^\[Mu]) /; Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 p z sin μ ( c z ) cosh v ( b + a z ) z 2 - v sin μ ( c z ) ( k = 0 v - 1 2 b ( v - 2 k ) ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p + a ( v - 2 k ) ) z p + a ( v - 2 k ) - c μ 2 F 1 ( - ( p + a ( v - 2 k ) - c μ ) 2 c , - μ ; 1 2 ( - ( p + a ( v - 2 k ) ) c - μ + 2 ) ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], ")"]]]], "c"]]], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + ( p - a ( v - 2 k ) ) z - 2 b ( v - 2 k ) p - a ( v - 2 k ) - c μ 2 F 1 ( - ( p - a ( v - 2 k ) - c μ ) 2 c , - μ ; 1 2 ( - ( p - a ( v - 2 k ) ) c - μ + 2 ) ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], ")"]]]], "c"]]], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) ( 1 - 2 c z ) - μ + 2 - v p z ( 1 - 2 c z ) - μ ( 1 - v mod 2 \$CellContext`v 2 ) sin μ ( c z ) p - c μ ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 2 F 1 ( - ( p - c μ ) 2 c , - μ ; 1 2 ( - p c - μ + 2 ) ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], "c"]]], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; v + Condition z p z c z μ b a z v 2 -1 v c z μ k 0 v -1 2 -1 b v -1 2 k Binomial v k p a v -1 2 k z p a v -1 2 k -1 c μ -1 Hypergeometric2F1 -1 p a v -1 2 k -1 c μ 2 c -1 -1 μ 1 2 -1 p a v -1 2 k c -1 -1 μ 2 2 c z p -1 a v -1 2 k z -1 2 b v -1 2 k p -1 a v -1 2 k -1 c μ -1 Hypergeometric2F1 -1 p -1 a v -1 2 k -1 c μ 2 c -1 -1 μ 1 2 -1 p -1 a v -1 2 k c -1 -1 μ 2 2 c z 1 -1 2 c z -1 μ 2 -1 v p z 1 -1 2 c z -1 μ 1 -1 \$CellContext`v 2 c z μ p -1 c μ -1 Binomial v v 2 -1 Hypergeometric2F1 -1 p -1 c μ 2 c -1 -1 μ 1 2 -1 p c -1 -1 μ 2 2 c z v SuperPlus [/itex]

 Rule Form

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

 2002-12-18