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 BesselK

 http://functions.wolfram.com/03.04.21.0032.01

 Input Form

 Integrate[Sinh[b + a z^r] BesselK[\[Nu], a z^r], z] == (2^(-1 - \[Nu]) Pi z Csc[Pi \[Nu]] (-((4^\[Nu] a z^r Cosh[b] HypergeometricPFQ[{3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 1/2 + 1/(2 r) - \[Nu]/2}, {3/2, 1 - \[Nu], 3/2 - \[Nu], 3/2 + 1/(2 r) - \[Nu]/2}, a^2 z^(2 r)])/((-1 + r (-1 + \[Nu])) Gamma[1 - \[Nu]])) - (1/((-1 + r \[Nu]) Gamma[1 - \[Nu]])) (4^\[Nu] HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/(2 r) - \[Nu]/2}, {1/2, 1/2 - \[Nu], 1 - \[Nu], 1 + 1/(2 r) - \[Nu]/2}, a^2 z^(2 r)] Sinh[b]) + (1/Gamma[1 + \[Nu]]) ((a z^r)^(2 \[Nu]) (-((1/(1 + r + r \[Nu])) (a z^r Cosh[b] HypergeometricPFQ[ {3/4 + \[Nu]/2, 5/4 + \[Nu]/2, 1/2 + 1/(2 r) + \[Nu]/2}, {3/2, 3/2 + 1/(2 r) + \[Nu]/2, 1 + \[Nu], 3/2 + \[Nu]}, a^2 z^(2 r)])) - (1/(1 + r \[Nu])) (HypergeometricPFQ[{1/4 + \[Nu]/2, 3/4 + \[Nu]/2, 1/(2 r) + \[Nu]/2}, {1/2, 1 + 1/(2 r) + \[Nu]/2, 1/2 + \[Nu], 1 + \[Nu]}, a^2 z^(2 r)] Sinh[b])))))/(a z^r)^\[Nu]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", "\[Pi]", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", "a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["Cosh", "[", "b", "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", 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RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]], " ", RowBox[List["Sinh", "[", "b", "]"]]]], ")"]]]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 sinh ( a z r + b ) K ν ( a z r ) z 2 - ν - 1 π z ( a z r ) - ν csc ( π ν ) ( - 4 ν sinh ( b ) ( r ν - 1 ) Γ ( 1 - ν ) 3 F 4 ( 1 4 - ν 2 , 3 4 - ν 2 , 1 2 r - ν 2 ; 1 2 , 1 2 - ν , 1 - ν , - ν 2 + 1 2 r + 1 ; a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 Γ ( ν + 1 ) ( ( a z r ) 2 ν ( - a z r cosh ( b ) ν r + r + 1 3 F 4 ( ν 2 + 3 4 , ν 2 + 5 4 , ν 2 + 1 2 + 1 2 r ; 3 2 , ν 2 + 3 2 + 1 2 r , ν + 1 , ν + 3 2 ; a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"], "+", FractionBox["1", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"], "+", FractionBox["1", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - sinh ( b ) r ν + 1 3 F 4 ( ν 2 + 1 4 , ν 2 + 3 4 , ν 2 + 1 2 r ; 1 2 , ν 2 + 1 2 r + 1 , ν + 1 2 , ν + 1 ; a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", RowBox[List["2", " ", "r"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) - 4 ν a z r cosh ( b ) ( r ( ν - 1 ) - 1 ) Γ ( 1 - ν ) 3 F 4 ( 3 4 - ν 2 , 5 4 - ν 2 , - ν 2 + 1 2 + 1 2 r ; 3 2 , 1 - ν , 3 2 - ν , - ν 2 + 3 2 + 1 2 r ; a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", FractionBox["1", "2"], "+", FractionBox["1", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "2"], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", FractionBox["3", "2"], "+", FractionBox["1", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) z a z r b BesselK ν a z r 2 -1 ν -1 z a z r -1 ν ν -1 4 ν b r ν -1 Gamma 1 -1 ν -1 HypergeometricPFQ 1 4 -1 ν 2 -1 3 4 -1 ν 2 -1 1 2 r -1 -1 ν 2 -1 1 2 1 2 -1 ν 1 -1 ν -1 ν 2 -1 1 2 r -1 1 a 2 z 2 r 1 Gamma ν 1 -1 a z r 2 ν -1 a z r b ν r r 1 -1 HypergeometricPFQ ν 2 -1 3 4 ν 2 -1 5 4 ν 2 -1 1 2 1 2 r -1 3 2 ν 2 -1 3 2 1 2 r -1 ν 1 ν 3 2 a 2 z 2 r -1 b r ν 1 -1 HypergeometricPFQ ν 2 -1 1 4 ν 2 -1 3 4 ν 2 -1 1 2 r -1 1 2 ν 2 -1 1 2 r -1 1 ν 1 2 ν 1 a 2 z 2 r -1 4 ν a z r b r ν -1 -1 Gamma 1 -1 ν -1 HypergeometricPFQ 3 4 -1 ν 2 -1 5 4 -1 ν 2 -1 -1 ν 2 -1 1 2 1 2 r -1 3 2 1 -1 ν 3 2 -1 ν -1 ν 2 -1 3 2 1 2 r -1 a 2 z 2 r [/itex]

 Rule Form

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

 2001-10-29