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Tanh




Mathematica Notation

Traditional Notation









Elementary Functions > Tanh[z] > Integration > Indefinite integration > Involving functions of the direct function and hyperbolic functions > Involving powers of the direct function and hyperbolic functions > Involving sinh and cosh





http://functions.wolfram.com/01.21.21.0347.01









  


  








Input Form





Integrate[Sinh[c z] Sqrt[a + b Cosh[2 c z]] Tanh[c z]^2, z] == (1/(4 c)) ((Sqrt[2] (a - 5 b) Log[Sqrt[2] Sqrt[b] Cosh[c z] + Sqrt[a + b Cosh[2 c z]]])/Sqrt[b] + (5 + Cosh[2 c z]) Sqrt[a + b Cosh[2 c z]] Sech[c z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", " ", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "c"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["5", " ", "b"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]], SqrtBox["b"]], "+", RowBox[List[RowBox[List["(", RowBox[List["5", "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]]]]]]








MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> b </mi> </msqrt> </mfrac> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["5", " ", "b"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]], SqrtBox["b"]], "+", RowBox[List[RowBox[List["(", RowBox[List["5", "+", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["4", " ", "c"]]]]]]]


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





2002-12-18