Exponential function: Integration (formula 01.03.21.0775)

Exp

$Mathematica Notation$

Elementary Functions

Exp[z]

Integration

Indefinite integration

Involving functions of the direct function and a power function

Involving rational functions of the direct function and a power function

Involving zⁿ(a+b c^{d z})^-m

http://functions.wolfram.com/01.03.21.0775.01

Input Form

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

Standard Form

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

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mfrac> <mi> z </mi> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> c </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ln /> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

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

2002-12-18