Integrate[(Sinh[a z]^(1/r))^b Cosh[a z], z] == (r Sinh[a z] (Sinh[a z]^(1/r))^b)/((r + b) a)

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], FractionBox["1", "r"]], ")"]], "b"], RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", FractionBox[RowBox[List["r", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], FractionBox["1", "r"]], ")"]], "b"]]], RowBox[List[RowBox[List["(", RowBox[List["r", "+", "b"]], ")"]], " ", "a"]]]]]]]

<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> <mrow> <msup> <mi> sinh </mi> <mfrac> <mn> 1 </mn> <mi> r </mi> </mfrac> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </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> <mfrac> <mrow> <mi> r </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> sinh </mi> <mfrac> <mn> 1 </mn> <mi> r </mi> </mfrac> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <apply> <cosh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <ci> r </ci> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> r </ci> </apply> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["a_", " ", "z_"]], "]"]], FractionBox["1", "r_"]], ")"]], "b_"], " ", RowBox[List["Cosh", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["r", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["1", "/", "r"]]], ")"]], "b"]]], RowBox[List[RowBox[List["(", RowBox[List["r", "+", "b"]], ")"]], " ", "a"]]]]]]]

2002-12-18