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 ArcSech

 http://functions.wolfram.com/01.30.21.0028.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Log[b z] ArcSech[a z], z] == (1/\[Alpha]^3) (z^\[Alpha] (\[Alpha] ArcSech[a z] (-1 + \[Alpha] Log[b z]) - (Sqrt[1 + a z] Sqrt[1 - a^2 z^2] HypergeometricPFQ[ {1/2, \[Alpha]/2, \[Alpha]/2}, {1 + \[Alpha]/2, 1 + \[Alpha]/2}, a^2 z^2] - ((1/2) Sqrt[1 + a z] Sqrt[1 - a^2 z^2] \[Alpha]^2 Beta[a^2 z^2, \[Alpha]/2, 1/2] Log[z])/(a^2 z^2)^(\[Alpha]/2) + ((1/2) Sqrt[1 - a z] (1 + a z) \[Alpha] Beta[a^2 z^2, \[Alpha]/2, 1/2] (1 + \[Alpha] Log[z] - \[Alpha] Log[b z]))/(a^2 z^2)^(\[Alpha]/2))/ (Sqrt[(1 - a z)/(1 + a z)] (1 + a z)^(3/2))))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["\[Alpha]", "3"]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Alpha]", " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", "2"], ",", FractionBox["\[Alpha]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SuperscriptBox["\[Alpha]", "2"], " ", RowBox[List["Beta", "[", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ",", FractionBox["\[Alpha]", "2"], ",", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", "\[Alpha]", " ", RowBox[List["Beta", "[", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ",", FractionBox["\[Alpha]", "2"], ",", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]], ")"]], RowBox[List["3", "/", "2"]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 log ( b z ) sech - 1 ( a z ) z z α α 3 ( α sech - 1 ( a z ) ( α log ( b z ) - 1 ) - ( - 1 2 a z + 1 1 - a 2 z 2 α 2 Β Beta a 2 z 2 ( α 2 , 1 2 ) log ( z ) ( a 2 z 2 ) - α 2 + 1 2 ( a z + 1 ) 1 - a z α Β Beta a 2 z 2 ( α 2 , 1 2 ) ( α log ( z ) - α log ( b z ) + 1 ) ( a 2 z 2 ) - α 2 + a z + 1 1 - a 2 z 2 3 F 2 ( 1 2 , α 2 , α 2 ; α 2 + 1 , α 2 + 1 ; a 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["\[Alpha]", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["\[Alpha]", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) / ( 1 - a z a z + 1 ( a z + 1 ) 3 / 2 ) ) z z α -1 b z a z z α α 3 -1 α a z α b z -1 -1 -1 1 2 a z 1 1 2 1 -1 a 2 z 2 1 2 α 2 Beta a 2 z 2 α 2 -1 1 2 z a 2 z 2 -1 α 2 -1 1 2 a z 1 1 -1 a z 1 2 α Beta a 2 z 2 α 2 -1 1 2 α z -1 α b z 1 a 2 z 2 -1 α 2 -1 a z 1 1 2 1 -1 a 2 z 2 1 2 HypergeometricPFQ 1 2 α 2 -1 α 2 -1 α 2 -1 1 α 2 -1 1 a 2 z 2 1 -1 a z a z 1 -1 1 2 a z 1 3 2 -1 [/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["Log", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["ArcSech", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Alpha]", " ", RowBox[List["ArcSech", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", "2"], ",", FractionBox["\[Alpha]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SuperscriptBox["\[Alpha]", "2"], " ", RowBox[List["Beta", "[", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ",", FractionBox["\[Alpha]", "2"], ",", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", "\[Alpha]", " ", RowBox[List["Beta", "[", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ",", FractionBox["\[Alpha]", "2"], ",", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "z"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], ")"]]]], SuperscriptBox["\[Alpha]", "3"]]]]]]

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

 2001-10-29