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 ArcCosh

 http://functions.wolfram.com/01.26.21.0007.01

 Input Form

 Integrate[ArcCosh[a z + b]/z, z] == ArcCosh[b + a z] Log[a z] + I ((-(I/2)) ArcCosh[b + a z]^2 - 4 ArcSin[Sqrt[1 - b]/Sqrt[2]] ArcTanh[((1 + b) Tanh[(1/2) ArcCosh[b + a z]])/Sqrt[-1 + b^2]] + ((-I) ArcCosh[b + a z] + 2 ArcSin[Sqrt[1 - b]/Sqrt[2]]) Log[1 + (-b + Sqrt[-1 + b^2])/E^ArcCosh[b + a z]] - (I ArcCosh[b + a z] + 2 ArcSin[Sqrt[1 - b]/Sqrt[2]]) Log[1 - (b + Sqrt[-1 + b^2])/E^ArcCosh[b + a z]] + I ArcCosh[b + a z] Log[a z] + I (PolyLog[2, -((-b + Sqrt[-1 + b^2])/E^ArcCosh[b + a z])] + PolyLog[2, (b + Sqrt[-1 + b^2])/E^ArcCosh[b + a z]]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcCosh", "[", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], "]"]], "z"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", "2"]]], SuperscriptBox[RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "b"]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["Tanh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "b"]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "b"]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]

 MathML Form

 cosh - 1 ( a z + b ) z z cosh - 1 ( b + a z ) log ( a z ) + ( - 2 cosh - 1 ( b + a z ) 2 + log ( a z ) cosh - 1 ( b + a z ) - 4 sin - 1 ( 1 - b 2 ) tanh - 1 ( b + 1 b 2 - 1 tanh ( 1 2 cosh - 1 ( b + a z ) ) ) + ( 2 sin - 1 ( 1 - b 2 ) - cosh - 1 ( b + a z ) ) log ( b 2 - 1 - b cosh - 1 ( b + a z ) + 1 ) - ( cosh - 1 ( b + a z ) + 2 sin - 1 ( 1 - b 2 ) ) log ( 1 - b + b 2 - 1 cosh - 1 ( b + a z ) ) + ( Li PolyLog 2 ( b + b 2 - 1 cosh - 1 ( b + a z ) ) + Li PolyLog 2 ( - b 2 - 1 - b cosh - 1 ( b + a z ) ) ) ) z a z b z -1 b a z a z -1 2 -1 b a z 2 a z b a z -1 4 1 -1 b 1 2 2 1 2 -1 b 1 b 2 -1 1 2 -1 1 2 b a z 2 1 -1 b 1 2 2 1 2 -1 -1 b a z b 2 -1 1 2 -1 b b a z -1 1 -1 b a z 2 1 -1 b 1 2 2 1 2 -1 1 -1 b b 2 -1 1 2 b a z -1 PolyLog 2 b b 2 -1 1 2 b a z -1 PolyLog 2 -1 b 2 -1 1 2 -1 b b a z -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["ArcCosh", "[", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], "]"]], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox[RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "b"]]], SqrtBox["2"]], "]"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["Tanh", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "b"]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "b"]]], SqrtBox["2"]], "]"]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]]]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["b", "+", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["b", "2"]]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["ArcCosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]

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

 2001-10-29