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 Cot

 http://functions.wolfram.com/01.09.21.0133.01

 Input Form

 Integrate[E^(p z) Sin[b z] Cot[c z], z] == (1/2) ((1/((-I) b + p)) (E^(I c z + ((-I) b - I c + p) z) HypergeometricPFQ[{1, -(b/(2 c)) - (I p)/(2 c)}, {1 - b/(2 c) - (I p)/(2 c)}, E^(2 I c z)]) + (1/((-I) b + 2 I c + p)) (E^(I c z + ((-I) b + I c + p) z) HypergeometricPFQ[{1, 1 - b/(2 c) - (I p)/(2 c)}, {2 - b/(2 c) - (I p)/(2 c)}, E^(2 I c z)]) - (1/(I b + p)) (E^(I c z + (I b - I c + p) z) HypergeometricPFQ[ {1, b/(2 c) - (I p)/(2 c)}, {1 + b/(2 c) - (I p)/(2 c)}, E^(2 I c z)]) - (1/(I b + 2 I c + p)) (E^(I c z + (I b + I c + p) z) HypergeometricPFQ[{1, 1 + b/(2 c) - (I p)/(2 c)}, {2 + b/(2 c) - (I p)/(2 c)}, E^(2 I c z)]))

 Standard Form

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 MathML Form

 p z sin ( b z ) cot ( c z ) z 1 2 ( - c z + ( b - c + p ) z 2 F 1 ( 1 , b 2 c - p 2 c ; b 2 c - p 2 c + 1 ; 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["1", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] b + p - c z + ( b + c + p ) z 2 F 1 ( 1 , b 2 c - p 2 c + 1 ; b 2 c - p 2 c + 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["1", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]], "+", "2"]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] b + 2 c + p + c z + ( - b - c + p ) z 2 F 1 ( 1 , - b 2 c - p 2 c ; - b 2 c - p 2 c + 1 ; 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["1", HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["b", RowBox[List["2", " ", "c"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox["b", RowBox[List["2", " ", "c"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - b + p + c z + ( - b + c + p ) z 2 F 1 ( 1 , - b 2 c - p 2 c + 1 ; - b 2 c - p 2 c + 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["1", HypergeometricPFQ], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["b", RowBox[List["2", " ", "c"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox["b", RowBox[List["2", " ", "c"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]], "+", "2"]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - b + 2 c + p ) z p z b z c z 1 2 -1 c z b -1 c p z HypergeometricPFQ 1 b 2 c -1 -1 p 2 c -1 b 2 c -1 -1 p 2 c -1 1 2 c z b p -1 -1 c z b c p z HypergeometricPFQ 1 b 2 c -1 -1 p 2 c -1 1 b 2 c -1 -1 p 2 c -1 2 2 c z b 2 c p -1 c z -1 b -1 c p z HypergeometricPFQ 1 -1 b 2 c -1 -1 p 2 c -1 -1 b 2 c -1 -1 p 2 c -1 1 2 c z -1 b p -1 c z -1 b c p z HypergeometricPFQ 1 -1 b 2 c -1 -1 p 2 c -1 1 -1 b 2 c -1 -1 p 2 c -1 2 2 c z -1 b 2 c p -1 [/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["Sin", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[RowBox[List["-", FractionBox["b", RowBox[List["2", " ", "c"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "p"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", "-", FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c"]], "+", "p"]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "p"]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", "+", FractionBox["b", RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c"]], "+", "p"]]]]], ")"]]]]]]]]

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

 2002-12-18