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http://functions.wolfram.com/01.28.21.0015.01
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Integrate[ArcCoth[a z^2 + b z + c], z] == z ArcCoth[a z^2 + b z + c] +
(Sqrt[-b^2 + 4 a + 4 a c] ArcTan[(b + 2 a z)/Sqrt[-b^2 + 4 a + 4 a c]])/
(2 a) + (b Log[a z^2 + b z + c + 1])/(4 a) -
(b Log[a z^2 + b z + c - 1])/(4 a) +
((b^2 + 4 a - 4 a c) ArcTan[(b + 2 a z)/Sqrt[-b^2 - 4 a + 4 a c]])/
(2 a Sqrt[-b^2 - 4 a + 4 a c])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcCoth", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcCoth", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]], "]"]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]]], "]"]]]], RowBox[List["2", " ", "a"]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c", "+", "1"]], "]"]]]], RowBox[List["4", " ", "a"]]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c", "-", "1"]], "]"]]]], RowBox[List["4", " ", "a"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["4", " ", "a"]], "-", RowBox[List["4", " ", "a", " ", "c"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "-", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]]], "]"]]]], RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "-", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <msqrt> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <arccoth /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> <ci> c </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <arccoth /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ln /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <ln /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["ArcCoth", "[", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["b_", " ", "z_"]], "+", "c_"]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", " ", RowBox[List["ArcCoth", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]], "]"]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "+", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]]], "]"]]]], RowBox[List["2", " ", "a"]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c", "+", "1"]], "]"]]]], RowBox[List["4", " ", "a"]]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c", "-", "1"]], "]"]]]], RowBox[List["4", " ", "a"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List["4", " ", "a"]], "-", RowBox[List["4", " ", "a", " ", "c"]]]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "-", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]]], "]"]]]], RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], "-", RowBox[List["4", " ", "a"]], "+", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
