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 ArcSinh

 http://functions.wolfram.com/01.25.21.0007.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcSinh", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", "8"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], "2"]]], "+", RowBox[List["32", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", "b"]], ")"]], " ", RowBox[List["Tan", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["b", "2"]]]]], "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], SqrtBox["2"]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["b", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], SqrtBox["2"]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["b", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["b", "2"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["b", "2"]]]]]], ")"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["ArcSinh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]

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

 2001-10-29