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 Exp

 http://functions.wolfram.com/01.03.21.0777.01

 Input Form

 Integrate[z^3/(a + b c^(d z))^2, z] == (1/(4 a^2)) (z^4 - (4 z^3)/(d Log[c]) + (4 a z^3)/(a d Log[c] + b c^(d z) d Log[c]) + (12 z^2 Log[1 + (b c^(d z))/a])/(d^2 Log[c]^2) - (4 z^3 Log[1 + (b c^(d z))/a])/(d Log[c]) - (12 z (-2 + d z Log[c]) PolyLog[2, -((b c^(d z))/a)])/(d^3 Log[c]^3) + (24 (-1 + d z Log[c]) PolyLog[3, -((b c^(d z))/a)])/(d^4 Log[c]^4) - (24 PolyLog[4, -((b c^(d z))/a)])/(d^4 Log[c]^4))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["z", "3"], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["a", "2"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "4"], "-", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "3"]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], "+", FractionBox[RowBox[List["4", " ", "a", " ", SuperscriptBox["z", "3"]]], RowBox[List[RowBox[List["a", " ", "d", " ", RowBox[List["Log", "[", "c", "]"]]]], "+", RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]], " ", "d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], "+", FractionBox[RowBox[List["12", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], "]"]]]], RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "2"]]]], "-", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "3"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], "-", FractionBox[RowBox[List["12", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["d", " ", "z", " ", RowBox[List["Log", "[", "c", "]"]]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], RowBox[List[SuperscriptBox["d", "3"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "3"]]]], "+", FractionBox[RowBox[List["24", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["d", " ", "z", " ", RowBox[List["Log", "[", "c", "]"]]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], RowBox[List[SuperscriptBox["d", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "4"]]]], "-", FractionBox[RowBox[List["24", " ", RowBox[List["PolyLog", "[", RowBox[List["4", ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], RowBox[List[SuperscriptBox["d", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "4"]]]]]], ")"]]]]]]]]

 MathML Form

 z 3 ( a + b c d z ) 2 z 1 4 a 2 ( z 4 - 4 log ( b c d z a + 1 ) z 3 d log ( c ) - 4 z 3 d log ( c ) + 4 a z 3 b d log ( c ) c d z + a d log ( c ) + 12 log ( b c d z a + 1 ) z 2 d 2 log 2 ( c ) - 12 ( d z log ( c ) - 2 ) Li PolyLog 2 ( - b c d z a ) z d 3 log 3 ( c ) + 24 ( d z log ( c ) - 1 ) Li PolyLog 3 ( - b c d z a ) d 4 log 4 ( c ) - 24 Li PolyLog 4 ( - b c d z a ) d 4 log 4 ( c ) ) z z 3 a b c d z 2 -1 1 4 a 2 -1 z 4 -1 4 b c d z a -1 1 z 3 d c -1 -1 4 z 3 d c -1 4 a z 3 b d c c d z a d c -1 12 b c d z a -1 1 z 2 d 2 c 2 -1 -1 12 d z c -2 PolyLog 2 -1 b c d z a -1 z d 3 c 3 -1 24 d z c -1 PolyLog 3 -1 b c d z a -1 d 4 c 4 -1 -1 24 PolyLog 4 -1 b c d z a -1 d 4 c 4 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["z_", "3"], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox["c_", RowBox[List["d_", " ", "z_"]]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "4"], "-", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "3"]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], "+", FractionBox[RowBox[List["4", " ", "a", " ", SuperscriptBox["z", "3"]]], RowBox[List[RowBox[List["a", " ", "d", " ", RowBox[List["Log", "[", "c", "]"]]]], "+", RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]], " ", "d", " ", RowBox[List["Log", "[", "c", "]"]]]]]]], "+", FractionBox[RowBox[List["12", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], "]"]]]], RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "2"]]]], "-", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "3"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]], "]"]]]], RowBox[List["d", " ", RowBox[List["Log", "[", "c", "]"]]]]], "-", FractionBox[RowBox[List["12", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["d", " ", "z", " ", RowBox[List["Log", "[", "c", "]"]]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], RowBox[List[SuperscriptBox["d", "3"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "3"]]]], "+", FractionBox[RowBox[List["24", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["d", " ", "z", " ", RowBox[List["Log", "[", "c", "]"]]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["3", ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], RowBox[List[SuperscriptBox["d", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "4"]]]], "-", FractionBox[RowBox[List["24", " ", RowBox[List["PolyLog", "[", RowBox[List["4", ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["c", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], RowBox[List[SuperscriptBox["d", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", "c", "]"]], "4"]]]]]], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]]]

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

 2002-12-18

© 1998-2013 Wolfram Research, Inc.