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Cosh






Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential function > Involving exp > Involving ab zr+d z cosh(c z)





http://functions.wolfram.com/01.20.21.0190.01









  


  










Input Form





Integrate[a^(b Sqrt[z] + d z) Cosh[c z], z] == (1/2) (b Sqrt[Pi] Log[a] (E^((b^2 Log[a]^2)/(4 (-c - d Log[a]))) (Erf[(-2 c Sqrt[z] - b Log[a] - 2 d Sqrt[z] Log[a])/ (2 Sqrt[-c - d Log[a]])]/(2 (-c - d Log[a])^(3/2))) + E^((b^2 Log[a]^2)/(4 (c - d Log[a]))) (Erf[(2 c Sqrt[z] - b Log[a] - 2 d Sqrt[z] Log[a])/ (2 Sqrt[c - d Log[a]])]/(2 (c - d Log[a])^(3/2)))) - (a^(b Sqrt[z] + d z) E^(c z))/(-c - d Log[a]) - a^(b Sqrt[z] + d z)/(E^(c z) (c - d Log[a])))










Standard Form





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MathML Form







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<apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> d </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> d </ci> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]]]]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Log", "[", "a", "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", SqrtBox["z"]]], "-", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "c", " ", SqrtBox["z"]]], "-", RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "a", "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]]], "]"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]]], ")"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]]]], RowBox[List["c", "-", RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]]]]]]], ")"]]]]]]]]










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





2002-12-18