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 BesselI

 http://functions.wolfram.com/03.02.21.0043.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["r", "+", "\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Cosh", "[", "b", "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]], " ", RowBox[List["Sinh", "[", "b", "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 cosh ( a z r + b ) I ν ( a z r ) z 1 ( α + r ν ) ( ν r + r + α ) Γ ( ν + 1 ) ( 2 - ν z α ( a z r ) ν ( a ( α + r ν ) 3 F 4 ( ν 2 + 3 4 , ν 2 + 5 4 , α 2 r + ν 2 + 1 2 ; 3 2 , α 2 r + ν 2 + 3 2 , 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["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], 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["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"]]], 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 ) z r + ( ν r + r + α ) cosh ( b ) 3 F 4 ( ν 2 + 1 4 , ν 2 + 3 4 , α 2 r + ν 2 ; 1 2 , α 2 r + ν 2 + 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["\[Alpha]", 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["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "+", "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] ) ) z z α -1 a z r b BesselI ν a z r 1 α r ν ν r r α Gamma ν 1 -1 2 -1 ν z α a z r ν a α r ν HypergeometricPFQ ν 2 -1 3 4 ν 2 -1 5 4 α 2 r -1 ν 2 -1 1 2 3 2 α 2 r -1 ν 2 -1 3 2 ν 1 ν 3 2 a 2 z 2 r b z r ν r r α b HypergeometricPFQ ν 2 -1 1 4 ν 2 -1 3 4 α 2 r -1 ν 2 -1 1 2 α 2 r -1 ν 2 -1 1 ν 1 2 ν 1 a 2 z 2 r [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["r", "+", "\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Cosh", "[", "b", "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]], " ", RowBox[List["Sinh", "[", "b", "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["r", "+", "\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]]]]

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

 2001-10-29