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 Cosh

 http://functions.wolfram.com/01.20.21.3864.01

 Input Form

 Integrate[E^(p z) Csc[b z] Cosh[c z]^v, z] == -((1/(b^2 - 2 I b p - p^2)) (2^(1 - v) E^((I b + p) z) (b - I p) Binomial[v, v/2] Hypergeometric2F1[(b - I p)/(2 b), 1, (3 b - I p)/(2 b), E^(2 I b z)] (1 - Mod[v, 2]))) - 2^(1 - v) Sum[(E^((I b + p - c (-2 s + v)) z) Binomial[v, s] ((b - I (p + c (-2 s + v))) Hypergeometric2F1[ (b - I p + I c (-2 s + v))/(2 b), 1, (3 b - I p + I c (-2 s + v))/ (2 b), E^(2 I b z)] + E^(2 c (-2 s + v) z) (b - I (p - c (-2 s + v))) Hypergeometric2F1[ (b - I (p + c (-2 s + v)))/(2 b), 1, (3 b - I (p + c (-2 s + v)))/ (2 b), E^(2 I b z)]))/(b^2 - 2 I b p - p^2 + c^2 (-2 s + v)^2), {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 p z csc ( b z ) cosh v ( c z ) z - 2 1 - v ( b + p ) z ( b - p ) ( 1 - v mod 2 \$CellContext`v 2 ) b 2 - 2 p b - p 2 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 F 1 ( b - p 2 b , 1 ; 3 b - p 2 b ; 2 b 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[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "b"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "b"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - 2 1 - v s = 0 v - 1 2 ( ( b + p - c ( v - 2 s ) ) z ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b - ( p + c ( v - 2 s ) ) ) 2 F 1 ( b - p + c ( v - 2 s ) 2 b , 1 ; 3 b - p + c ( v - 2 s ) 2 b ; 2 b 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[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["c", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "b"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["c", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "b"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 c ( v - 2 s ) z ( b - ( p - c ( v - 2 s ) ) ) 2 F 1 ( b - ( p + c ( v - 2 s ) ) 2 b , 1 ; 3 b - ( p + c ( v - 2 s ) ) 2 b ; 2 b 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[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]]]]]], RowBox[List["2", " ", "b"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]]]]]], RowBox[List["2", " ", "b"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) / ( b 2 - 2 p b - p 2 + c 2 ( v - 2 s ) 2 ) /; v + Condition z p z b z c z v -1 2 1 -1 v b p z b -1 p 1 -1 \$CellContext`v 2 b 2 -1 2 p b -1 p 2 -1 Binomial v v 2 -1 Hypergeometric2F1 b -1 p 2 b -1 1 3 b -1 p 2 b -1 2 b z -1 2 1 -1 v s 0 v -1 2 -1 b p -1 c v -1 2 s z Binomial v s b -1 p c v -1 2 s Hypergeometric2F1 b -1 p c v -1 2 s 2 b -1 1 3 b -1 p c v -1 2 s 2 b -1 2 b z 2 c v -1 2 s z b -1 p -1 c v -1 2 s Hypergeometric2F1 b -1 p c v -1 2 s 2 b -1 1 3 b -1 p c v -1 2 s 2 b -1 2 b z b 2 -1 2 p b -1 p 2 c 2 v -1 2 s 2 -1 v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Csc", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "b"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "b"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "p"]], "-", SuperscriptBox["p", "2"]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "v"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], RowBox[List["2", " ", "b"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], RowBox[List["2", " ", "b"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], RowBox[List["2", " ", "b"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]]]], RowBox[List["2", " ", "b"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]]]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "p"]], "-", SuperscriptBox["p", "2"], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18