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 Log

 http://functions.wolfram.com/01.04.21.0029.01

 Input Form

 Integrate[Log[Sin[z] + a], z] == (1/8) ((-I) (Pi - 2 z)^2 + 32 I ArcSin[Sqrt[1 + a]/Sqrt[2]] ArcTan[((-1 + a) Tan[(1/4) (Pi - 2 z)])/Sqrt[-1 + a^2]] + 4 (Pi - 2 z + 4 ArcSin[Sqrt[1 + a]/Sqrt[2]]) Log[1 + (I (a - Sqrt[-1 + a^2]))/E^(I z)] + 4 (Pi - 2 z - 4 ArcSin[Sqrt[1 + a]/Sqrt[2]]) Log[1 + (I (a + Sqrt[-1 + a^2]))/E^(I z)] - 4 (Pi - 2 z) Log[a + Sin[z]] - 8 I (PolyLog[2, (I (-a + Sqrt[-1 + a^2]))/E^(I z)] + PolyLog[2, ((-I) (a + Sqrt[-1 + a^2]))/E^(I z)]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Log", "[", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "+", "a"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "z"]]]], ")"]], "2"]]], "+", RowBox[List["32", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "+", "a"]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", RowBox[List["Tan", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "z"]]]], ")"]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["a", "2"]]]]], "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "z"]], "+", RowBox[List["4", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "+", "a"]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["a", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]]]], "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "z"]], "-", RowBox[List["4", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "+", "a"]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["a", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]]]], "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["Sin", "[", "z", "]"]]]], "]"]]]], "-", RowBox[List["8", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["a", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["a", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 log ( a + sin ( z ) ) z 1 8 ( - ( π - 2 z ) 2 - 4 log ( a + sin ( z ) ) ( π - 2 z ) + 32 sin - 1 ( a + 1 2 ) tan - 1 ( ( a - 1 ) tan ( 1 4 ( π - 2 z ) ) a 2 - 1 ) + 4 ( - 2 z + 4 sin - 1 ( a + 1 2 ) + π ) log ( 1 + ( a - a 2 - 1 ) - z ) + 4 ( - 2 z - 4 sin - 1 ( a + 1 2 ) + π ) log ( 1 + ( a + a 2 - 1 ) - z ) - 8 ( Li PolyLog 2 ( ( a 2 - 1 - a ) - z ) + Li PolyLog 2 ( - ( a + a 2 - 1 ) - z ) ) ) z a z 1 8 -1 -1 2 z 2 -1 4 a z -1 2 z 32 a 1 1 2 2 1 2 -1 a -1 1 4 -1 2 z a 2 -1 1 2 -1 4 -2 z 4 a 1 1 2 2 1 2 -1 1 a -1 a 2 -1 1 2 -1 z 4 -2 z -1 4 a 1 1 2 2 1 2 -1 1 a a 2 -1 1 2 -1 z -1 8 PolyLog 2 a 2 -1 1 2 -1 a -1 z PolyLog 2 -1 a a 2 -1 1 2 -1 z [/itex]

 Rule Form

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

 2001-10-29