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 Cos

 http://functions.wolfram.com/01.07.21.2576.01

 Input Form

 Integrate[(z Sin[2 c z])/(a Sin[c z] + b Cos[c z]), z] == (2/c^2) ((2 a b ArcTan[b/a] ArcTanh[(-a + b Tan[(c z)/2])/Sqrt[a^2 + b^2]])/ (a^2 + b^2)^(3/2) + ((a - b c z) Cos[c z])/(a^2 + b^2) - (b ((c z + ArcTan[b/a]) (Log[1 - E^(I (c z + ArcTan[b/a]))] - Log[1 + E^(I (c z + ArcTan[b/a]))]) + I (PolyLog[2, -E^(I (c z + ArcTan[b/a]))] - PolyLog[2, E^(I (c z + ArcTan[b/a]))])))/ ((a^2 + b^2) Sqrt[1 + b^2/a^2]) + ((b + a c z) Sin[c z])/(a^2 + b^2))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], RowBox[List[RowBox[List["a", " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["2", SuperscriptBox["c", "2"]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", "a", " ", "b", " ", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["b", " ", RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["b", " ", "c", " ", "z"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "c", " ", "z"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]]]]]]

 MathML Form

 z sin ( 2 c z ) a sin ( c z ) + b cos ( c z ) z 2 c 2 ( 2 a b tan - 1 ( b a ) ( a 2 + b 2 ) 3 / 2 tanh - 1 ( b tan ( c z 2 ) - a a 2 + b 2 ) + ( a - b c z ) cos ( c z ) a 2 + b 2 + ( b + a c z ) sin ( c z ) a 2 + b 2 - 1 ( a 2 + b 2 ) b 2 a 2 + 1 ( b ( ( c z + tan - 1 ( b a ) ) ( log ( 1 - ( c z + tan - 1 ( b a ) ) ) - log ( 1 + ( c z + tan - 1 ( b a ) ) ) ) + ( Li PolyLog 2 ( - ( c z + tan - 1 ( b a ) ) ) - Li PolyLog 2 ( ( c z + tan - 1 ( b a ) ) ) ) ) ) ) z z 2 c z a c z b c z -1 2 c 2 -1 2 a b b a -1 a 2 b 2 3 2 -1 b c z 2 -1 -1 a a 2 b 2 1 2 -1 a -1 b c z c z a 2 b 2 -1 b a c z c z a 2 b 2 -1 -1 1 a 2 b 2 b 2 a 2 -1 1 1 2 -1 b c z b a -1 1 -1 c z b a -1 -1 1 c z b a -1 PolyLog 2 -1 c z b a -1 -1 PolyLog 2 c z b a -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["z_", " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]], RowBox[List[RowBox[List["a_", " ", RowBox[List["Sin", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", "a", " ", "b", " ", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["b", " ", RowBox[List["Tan", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["b", " ", "c", " ", "z"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", "z"]], "+", RowBox[List["ArcTan", "[", FractionBox["b", "a"], "]"]]]], ")"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SqrtBox[RowBox[List["1", "+", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["a", "2"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "c", " ", "z"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]]], ")"]]]], SuperscriptBox["c", "2"]]]]]]

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

 2002-12-18