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 Cosh

 http://functions.wolfram.com/01.20.21.0149.01

 Input Form

 Integrate[(z^4 Cosh[c z])/(a z + b)^2, z] == (1/(a^6 c^3 (b + a z))) ((-b^3) c^3 (b + a z) CoshIntegral[c (b/a + z)] (4 a Cosh[(b c)/a] + b c Sinh[(b c)/a]) + a ((-c) (-2 a^2 b^2 + b^4 c^2 + 2 a^4 z^2) Cosh[c z] + a (b + a z) (3 b^2 c^2 - 2 a b c^2 z + a^2 (2 + c^2 z^2)) Sinh[c z]) + b^3 c^3 (b + a z) (b c Cosh[(b c)/a] + 4 a Sinh[(b c)/a]) SinhIntegral[c (b/a + z)])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "4"], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["a", "6"], " ", SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["b", "3"]]], " ", SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "z"]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]], "+", RowBox[List["b", " ", "c", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]]]], ")"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["2", " ", "a", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "c", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]], "+", RowBox[List["4", " ", "a", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "z"]], ")"]]]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 z 4 cosh ( c z ) ( b + a z ) 2 z 1 a 6 c 3 ( b + a z ) ( - b 3 ( b + a z ) Chi ( c ( b a + z ) ) ( 4 a cosh ( b c a ) + b c sinh ( b c a ) ) c 3 + b 3 ( b + a z ) ( b c cosh ( b c a ) + 4 a sinh ( b c a ) ) Shi ( c ( b a + z ) ) c 3 + a ( a ( b + a z ) ( ( c 2 z 2 + 2 ) a 2 - 2 b c 2 z a + 3 b 2 c 2 ) sinh ( c z ) - c ( 2 z 2 a 4 - 2 b 2 a 2 + b 4 c 2 ) cosh ( c z ) ) ) z z 4 c z b a z 2 -1 1 a 6 c 3 b a z -1 -1 b 3 b a z CoshIntegral c b a -1 z 4 a b c a -1 b c b c a -1 c 3 b 3 b a z b c b c a -1 4 a b c a -1 SinhIntegral c b a -1 z c 3 a a b a z c 2 z 2 2 a 2 -1 2 b c 2 z a 3 b 2 c 2 c z -1 c 2 z 2 a 4 -1 2 b 2 a 2 b 4 c 2 c z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z_", "4"], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["b", "3"]]], " ", SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "z"]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]], "+", RowBox[List["b", " ", "c", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]]]], ")"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["c", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["c", "2"]]], "-", RowBox[List["2", " ", "a", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "c", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]], "+", RowBox[List["4", " ", "a", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["b", " ", "c"]], "a"], "]"]]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["c", " ", RowBox[List["(", RowBox[List[FractionBox["b", "a"], "+", "z"]], ")"]]]], "]"]]]]]], RowBox[List[SuperscriptBox["a", "6"], " ", SuperscriptBox["c", "3"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]]]]

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

 2002-12-18