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

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Tanh






Mathematica Notation

Traditional Notation









Elementary Functions > Tanh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function > Involving exp > Involving ab z





http://functions.wolfram.com/01.21.21.0023.01









  


  










Input Form





Integrate[a^(b z) Tanh[c z], z] == (a^(b z) (b E^(2 c z) Hypergeometric2F1[1 + (b Log[a])/(2 c), 1, 2 + (b Log[a])/(2 c), -E^(2 c z)] Log[a] - Hypergeometric2F1[(b Log[a])/(2 c), 1, 1 + (b Log[a])/(2 c), -E^(2 c z)] (2 c + b Log[a])))/(b Log[a] (2 c + b Log[a]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["a", RowBox[List["b", " ", "z"]]], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], "-", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]], ")"]]]]]]]]










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> <mrow> <msup> <mi> a </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <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> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;a&quot;, &quot;)&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;a&quot;, &quot;)&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], &quot;+&quot;, &quot;2&quot;]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;a&quot;, &quot;)&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], Hypergeometric2F1], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List[&quot;b&quot;, &quot; &quot;, RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;a&quot;, &quot;)&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;]]], &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, SuperscriptBox[&quot;\[ExponentialE]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]], Hypergeometric2F1]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </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 /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ln /> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Hypergeometric2F1 </ci> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> a </ci> </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[RowBox[List[SuperscriptBox["a_", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]], " ", RowBox[List["Log", "[", "a", "]"]]]], "-", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]]]]]










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





2002-12-18





© 1998-2014 Wolfram Research, Inc.