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http://functions.wolfram.com/01.07.21.1262.01
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Integrate[(A + B Cos[c z])/(a + b Cos[c z])^3, z] ==
(1/(2 (a^2 - b^2)^2 c))
(-((2 (2 a^2 A + A b^2 - 3 a b B) ArcTanh[((a - b) Tan[(c z)/2])/
Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2]) +
((a^2 - b^2) ((-A) b + a B) Sin[c z])/(a + b Cos[c z])^2 +
((-3 a A b + a^2 B + 2 b^2 B) Sin[c z])/(a + b Cos[c z]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "2"], " ", "c"]]], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "A"]], "+", RowBox[List["A", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["3", " ", "a", " ", "b", " ", "B"]]]], ")"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "b"]], "+", RowBox[List["a", " ", "B"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "a", " ", "A", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "B"]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]
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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> <mfrac> <mrow> <mi> A </mi> <mo> + </mo> <mrow> <mi> B </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> A </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> B </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> A </mi> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> tan </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> B </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> A </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> B </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> B </mi> </mrow> <mo> - </mo> <mrow> <mi> A </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <plus /> <ci> A </ci> <apply> <times /> <ci> B </ci> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> A </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> <ci> B </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <ci> A </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <arctanh /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <tan /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> B </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> A </ci> <ci> b </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> B </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> B </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> A </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cos /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </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[FractionBox[RowBox[List["A_", "+", RowBox[List["B_", " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "3"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "A"]], "+", RowBox[List["A", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List["3", " ", "a", " ", "b", " ", "B"]]]], ")"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "A"]], " ", "b"]], "+", RowBox[List["a", " ", "B"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "a", " ", "A", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "B"]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", "B"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "2"], " ", "c"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
