Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving rational functions of the direct function > Involving ed z/a+b ec z





http://functions.wolfram.com/01.03.21.0434.01









  


  










Input Form





Integrate[E^(c z)/(a + b E^(3 c z)), z] == (-(1/(6 a^(2/3) b^(1/3) c))) (2 Sqrt[3] ArcTan[(1 - (2 b^(1/3) E^(c z))/a^(1/3))/Sqrt[3]] - 2 Log[a^(1/3) + b^(1/3) E^(c z)] + Log[a^(2/3) - a^(1/3) b^(1/3) E^(c z) + b^(2/3) E^(2 c z)])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["3", " ", "c", " ", "z"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["6", " ", SuperscriptBox["a", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "c"]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["3"], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["1", "-", FractionBox[RowBox[List["2", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], SuperscriptBox["a", RowBox[List["1", "/", "3"]]]]]], SqrtBox["3"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["a", RowBox[List["1", "/", "3"]]], "+", RowBox[List[SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["a", RowBox[List["2", "/", "3"]]], "-", RowBox[List[SuperscriptBox["a", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], "+", RowBox[List[SuperscriptBox["b", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]]]]]]










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> &#8747; </mo> <mrow> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mroot> <mi> a </mi> <mn> 3 </mn> </mroot> </mfrac> </mrow> <msqrt> <mn> 3 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mroot> <mi> a </mi> <mn> 3 </mn> </mroot> <mo> + </mo> <mrow> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mroot> <mi> a </mi> <mn> 3 </mn> </mroot> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mroot> <mi> b </mi> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <msup> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 2 <sep /> 3 </cn> </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[SuperscriptBox["\[ExponentialE]", RowBox[List["c_", " ", "z_"]]], RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["3", " ", "c_", " ", "z_"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox["3"], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["1", "-", FractionBox[RowBox[List["2", " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], SuperscriptBox["a", RowBox[List["1", "/", "3"]]]]]], SqrtBox["3"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["a", RowBox[List["1", "/", "3"]]], "+", RowBox[List[SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["a", RowBox[List["2", "/", "3"]]], "-", RowBox[List[SuperscriptBox["a", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], "+", RowBox[List[SuperscriptBox["b", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["6", " ", SuperscriptBox["a", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["b", RowBox[List["1", "/", "3"]]], " ", "c"]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.