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 ArcCsch

 http://functions.wolfram.com/01.29.21.0027.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Log[b z] ArcCsch[a z], z] == (1/\[Alpha]^3) (z^\[Alpha] (\[Alpha] ArcCsch[a z] (-1 + \[Alpha] Log[b z]) - (1/(a Sqrt[1 + 1/(a^2 z^2)] z)) (Sqrt[1 + a^2 z^2] (HypergeometricPFQ[{1/2, \[Alpha]/2, \[Alpha]/2}, {1 + \[Alpha]/2, 1 + \[Alpha]/2}, (-a^2) z^2] - \[Alpha] Hypergeometric2F1[1/2, \[Alpha]/2, 1 + \[Alpha]/2, (-a^2) z^2] Log[z] + Hypergeometric2F1[\[Alpha]/2, 1/2, 1 + \[Alpha]/2, (-a^2) z^2] (1 + \[Alpha] Log[z] - \[Alpha] Log[b z])))))

 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["ArcCsch", "[", 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["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["a", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]], " ", "z"]]], RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["(", RowBox[List[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[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], "-", RowBox[List["\[Alpha]", " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 log ( b z ) csch - 1 ( a z ) z z α α 3 ( α csch - 1 ( a z ) ( α log ( b z ) - 1 ) - 1 a 1 + 1 a 2 z 2 z ( a 2 z 2 + 1 ( 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[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] - α 2 F 1 ( 1 2 , α 2 ; α 2 + 1 ; - a 2 z 2 ) 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[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox["\[Alpha]", "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] log ( z ) + 2 F 1 ( α 2 , 1 2 ; α 2 + 1 ; - a 2 z 2 ) 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[FractionBox["\[Alpha]", "2"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ( α log ( z ) - α log ( b z ) + 1 ) ) ) ) z z α -1 b z a z z α α 3 -1 α a z α b z -1 -1 1 a 1 1 a 2 z 2 -1 1 2 z -1 a 2 z 2 1 1 2 HypergeometricPFQ 1 2 α 2 -1 α 2 -1 α 2 -1 1 α 2 -1 1 -1 a 2 z 2 -1 α Hypergeometric2F1 1 2 α 2 -1 α 2 -1 1 -1 a 2 z 2 z Hypergeometric2F1 α 2 -1 1 2 α 2 -1 1 -1 a 2 z 2 α z -1 α b z 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["ArcCsch", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Alpha]", " ", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["(", RowBox[List[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[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], "-", RowBox[List["\[Alpha]", " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["\[Alpha]", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "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["a", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]], " ", "z"]]]]], ")"]]]], SuperscriptBox["\[Alpha]", "3"]]]]]]

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

 2001-10-29