html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cot

 http://functions.wolfram.com/01.09.21.0035.01

 Input Form

 Integrate[Cot[a ArcSinh[z]], z] == (-(1/(2 (1 + 4 a^2)))) ((I ((1 - 2 I a) E^(2 (1 + I a) ArcSinh[z]) Hypergeometric2F1[1 - I/(2 a), 1, 2 - I/(2 a), E^(2 I a ArcSinh[z])] + (-I + 2 a) ((-I) E^(2 I a ArcSinh[z]) Hypergeometric2F1[1 + I/(2 a), 1, 2 + I/(2 a), E^(2 I a ArcSinh[z])] + (I + 2 a) (E^(2 ArcSinh[z]) Hypergeometric2F1[-(I/(2 a)), 1, 1 - I/(2 a), E^(2 I a ArcSinh[z])] - Hypergeometric2F1[I/(2 a), 1, 1 + I/(2 a), E^(2 I a ArcSinh[z])]))))/E^ArcSinh[z])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cot", "[", RowBox[List["a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], ")"]]]]]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]], " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 cot ( a sinh - 1 ( z ) ) z - 1 2 ( 4 a 2 + 1 ) ( - sinh - 1 ( z ) ( ( 2 a - ) ( ( 2 a + ) ( 2 sinh - 1 ( z ) 2 F 1 ( - 2 a , 1 ; 1 - 2 a ; 2 a sinh - 1 ( z ) ) 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[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", RowBox[List["-", "1"]]], "(", "z", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 2 F 1 ( 2 a , 1 ; 1 + 2 a ; 2 a sinh - 1 ( z ) ) 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["\[ImaginaryI]", RowBox[List["2", " ", "a"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", RowBox[List["-", "1"]]], "(", "z", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) - 2 a sinh - 1 ( z ) 2 F 1 ( 1 + 2 a , 1 ; 2 + 2 a ; 2 a sinh - 1 ( z ) ) 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[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", RowBox[List["-", "1"]]], "(", "z", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) + ( 1 - 2 a ) 2 ( 1 + a ) sinh - 1 ( z ) 2 F 1 ( 1 - 2 a , 1 ; 2 - 2 a ; 2 a sinh - 1 ( z ) ) 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[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List[SuperscriptBox["sinh", RowBox[List["-", "1"]]], "(", "z", ")"]]]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) z a z -1 1 2 4 a 2 1 -1 -1 z 2 a -1 2 a 2 z Hypergeometric2F1 -1 2 a -1 1 1 -1 2 a -1 2 a z -1 Hypergeometric2F1 2 a -1 1 1 2 a -1 2 a z -1 2 a z Hypergeometric2F1 1 2 a -1 1 2 2 a -1 2 a z 1 -1 2 a 2 1 a z Hypergeometric2F1 1 -1 2 a -1 1 2 -1 2 a -1 2 a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cot", "[", RowBox[List["a_", " ", RowBox[List["ArcSinh", "[", "z_", "]"]]]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "a"]]]], ")"]], " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["2", " ", "a"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", "1", ",", RowBox[List["1", "-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], "-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "a"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], ")"]]]]]]]]]]]

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

 2002-12-18