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 AiryAi

 http://functions.wolfram.com/03.05.21.0027.01

 Input Form

 Integrate[Cosh[b + (2/3) (a z^r)^(3/2)] AiryAi[a z^r], z] == (z (3 (1 + r) Gamma[4/3] HypergeometricPFQ[{1/6, 2/(3 r)}, {1/3, 1 + 2/(3 r)}, (-(4/3)) (a z^r)^(3/2)] + 3 E^(2 b) (1 + r) Gamma[4/3] HypergeometricPFQ[{1/6, 2/(3 r)}, {1/3, 1 + 2/(3 r)}, (4/3) (a z^r)^(3/2)] - 3^(1/3) a z^r Gamma[2/3] (HypergeometricPFQ[{5/6, 2/3 + 2/(3 r)}, {5/3, 5/3 + 2/(3 r)}, (-(4/3)) (a z^r)^(3/2)] + E^(2 b) HypergeometricPFQ[ {5/6, 2/3 + 2/(3 r)}, {5/3, 5/3 + 2/(3 r)}, (4/3) (a z^r)^(3/2)])))/ E^b/(6 3^(2/3) (1 + r) Gamma[2/3] Gamma[4/3])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], RowBox[List["AiryAi", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "+", "r"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", RowBox[List["3", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", RowBox[List["1", "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["(", RowBox[List["1", "+", "r"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", RowBox[List["3", " ", "r"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", RowBox[List["1", "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", RowBox[List[FractionBox["2", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "3"], ",", RowBox[List[FractionBox["5", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", RowBox[List[FractionBox["2", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "3"], ",", RowBox[List[FractionBox["5", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["6", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["1", "+", "r"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]]]], ")"]]]]]]]]

 MathML Form

 cosh ( 2 3 ( a z r ) 3 / 2 + b ) Ai ( a z r ) z 1 6 3 2 / 3 ( r + 1 ) Γ ( 2 3 ) Γ ( 4 3 ) ( - b z ( - 3 3 a Γ ( 2 3 ) ( 2 b 2 F 2 ( 5 6 , 2 3 + 2 3 r ; 5 3 , 5 3 + 2 3 r ; 4 3 ( a z r ) 3 / 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["2", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["5", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 2 F 2 ( 5 6 , 2 3 + 2 3 r ; 5 3 , 5 3 + 2 3 r ; 1 3 ( - 4 ) ( a z r ) 3 / 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["2", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["5", "3"], "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) z r + 3 2 b ( r + 1 ) Γ ( 4 3 ) 2 F 2 ( 1 6 , 2 3 r ; 1 3 , 1 + 2 3 r ; 4 3 ( a z r ) 3 / 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["2", RowBox[List["3", " ", "r"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 3 ( r + 1 ) Γ ( 4 3 ) 2 F 2 ( 1 6 , 2 3 r ; 1 3 , 1 + 2 3 r ; 1 3 ( - 4 ) ( a z r ) 3 / 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["2", RowBox[List["3", " ", "r"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", FractionBox["2", RowBox[List["3", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) z 2 3 a z r 3 2 b AiryAi a z r 1 6 3 2 3 r 1 Gamma 2 3 Gamma 4 3 -1 -1 b z -1 3 1 3 a Gamma 2 3 2 b HypergeometricPFQ 5 6 2 3 2 3 r -1 5 3 5 3 2 3 r -1 4 3 a z r 3 2 HypergeometricPFQ 5 6 2 3 2 3 r -1 5 3 5 3 2 3 r -1 1 3 -4 a z r 3 2 z r 3 2 b r 1 Gamma 4 3 HypergeometricPFQ 1 6 2 3 r -1 1 3 1 2 3 r -1 4 3 a z r 3 2 3 r 1 Gamma 4 3 HypergeometricPFQ 1 6 2 3 r -1 1 3 1 2 3 r -1 1 3 -4 a z r 3 2 [/itex]

 Rule Form

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

 2001-10-29