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 Exp

 http://functions.wolfram.com/01.03.21.0789.01

 Input Form

 Integrate[z/(a E^(2 d z) + b E^(d z) + c), z] == -(d z ((-Sqrt[b^2 - 4 a c]) d z + (b + Sqrt[b^2 - 4 a c]) Log[1 + (2 a E^(d z))/(b - Sqrt[b^2 - 4 a c])] + (-b + Sqrt[b^2 - 4 a c]) Log[1 + (2 a E^(d z))/ (b + Sqrt[b^2 - 4 a c])]) + (b + Sqrt[b^2 - 4 a c]) PolyLog[2, (2 a E^(d z))/(-b + Sqrt[b^2 - 4 a c])] + (-b + Sqrt[b^2 - 4 a c]) PolyLog[2, -((2 a E^(d z))/(b + Sqrt[b^2 - 4 a c]))])/(2 c Sqrt[b^2 - 4 a c] d^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox["z", RowBox[List[RowBox[List["a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", "d", " ", "z"]]]]], "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "+", "c"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], " ", "d", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], RowBox[List["b", "-", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], RowBox[List["b", "+", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]]]]]]]]], "]"]]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["2", " ", "c", " ", SqrtBox[RowBox[List[SuperscriptBox["b", "2"], "-", RowBox[List["4", " ", "a", " ", "c"]]]]], " ", SuperscriptBox["d", "2"]]], ")"]]]]]]]]

 MathML Form

 z a 2 d z + b d z + c z - 1 2 c b 2 - 4 a c d 2 ( d z ( - b 2 - 4 a c d z + ( b + b 2 - 4 a c ) log ( 2 d z a b - b 2 - 4 a c + 1 ) + ( b 2 - 4 a c - b ) log ( 2 d z a b + b 2 - 4 a c + 1 ) ) + ( b + b 2 - 4 a c ) Li PolyLog 2 ( 2 a d z b 2 - 4 a c - b ) + ( b 2 - 4 a c - b ) Li PolyLog 2 ( - 2 a d z b + b 2 - 4 a c ) ) z z a 2 d z b d z c -1 -1 1 2 c b 2 -1 4 a c 1 2 d 2 -1 d z -1 b 2 -1 4 a c 1 2 d z b b 2 -1 4 a c 1 2 2 d z a b -1 b 2 -1 4 a c 1 2 -1 1 b 2 -1 4 a c 1 2 -1 b 2 d z a b b 2 -1 4 a c 1 2 -1 1 b b 2 -1 4 a c 1 2 PolyLog 2 2 a d z b 2 -1 4 a c 1 2 -1 b -1 b 2 -1 4 a c 1 2 -1 b PolyLog 2 -1 2 a d z b b 2 -1 4 a c 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18