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 ArcCsch

 http://functions.wolfram.com/01.29.21.0031.01

 Input Form

 Integrate[z^(\[Alpha] - 1) ArcCsch[a z]^2, z] == (1/(2 a^2 (-1 + \[Alpha]) \[Alpha])) (z^(-2 + \[Alpha]) (2 a z ArcCsch[a z] (a z (-1 + \[Alpha]) ArcCsch[a z] + 2 Sqrt[1 + 1/(a^2 z^2)] Hypergeometric2F1[1, 1 - \[Alpha]/2, (3 - \[Alpha])/2, -(1/(a^2 z^2))]) + 2^\[Alpha] Sqrt[Pi] (-1 + \[Alpha]) Gamma[1 - \[Alpha]] HypergeometricPFQRegularized[ {1, 1 - \[Alpha]/2, 1 - \[Alpha]/2}, {(3 - \[Alpha])/2, 2 - \[Alpha]/2}, -(1/(a^2 z^2))]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]], ")"]], " ", "\[Alpha]"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "+", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "z", " ", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"], ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"], ",", RowBox[List["2", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 csch - 1 ( a z ) 2 z 1 2 a 2 ( α - 1 ) α z α - 2 ( 2 a z csch - 1 ( a z ) ( a z ( α - 1 ) csch - 1 ( a z ) + 2 1 + 1 a 2 z 2 2 F 1 ( 1 , 1 - α 2 ; 3 - 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["1", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", 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] ) + 2 α π ( α - 1 ) Γ ( 1 - α ) 3 F ~ 2 ( 1 , 1 - α 2 , 1 - α 2 ; 3 - α 2 , 2 - α 2 ; - 1 a 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) z z α -1 a z 2 1 2 a 2 α -1 α -1 z α -2 2 a z a z a z α -1 a z 2 1 1 a 2 z 2 -1 1 2 Hypergeometric2F1 1 1 -1 α 2 -1 3 -1 α 2 -1 -1 1 a 2 z 2 -1 2 α 1 2 α -1 Gamma 1 -1 α HypergeometricPFQRegularized 1 1 -1 α 2 -1 1 -1 α 2 -1 3 -1 α 2 -1 2 -1 α 2 -1 -1 1 a 2 z 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"]]], " ", SuperscriptBox[RowBox[List["ArcCsch", "[", RowBox[List["a_", " ", "z_"]], "]"]], "2"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "+", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "z", " ", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["ArcCsch", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"], ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"], ",", RowBox[List["2", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]], ")"]], " ", "\[Alpha]"]]]]]]]

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

 2001-10-29