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 Cosh

 http://functions.wolfram.com/01.20.21.1428.01

 Input Form

 Integrate[z^(\[Alpha] - 1) E^(p z) Cos[d + c z]^m Cosh[a z], z] == 2^(-1 - m) z^\[Alpha] (Binomial[m, m/2] (ExpIntegralE[1 - \[Alpha], (a - p) z] + ExpIntegralE[1 - \[Alpha], (-(a + p)) z]) (-1 + Mod[m, 2]) - Sum[(Binomial[m, k] (E^(4 I d k) (ExpIntegralE[1 - \[Alpha], (a + I c (-2 k + m) - p) z] + ExpIntegralE[1 - \[Alpha], (-(a + 2 I c k - I c m + p)) z]) + E^(2 I d m) (ExpIntegralE[1 - \[Alpha], (a + 2 I c k - I c m - p) z] + ExpIntegralE[1 - \[Alpha], (-(a - 2 I c k + I c m + p)) z])))/ E^(I d (2 k + m)), {k, 0, Floor[(1/2) (-1 + m)]}]) /; Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "p"]], ")"]], " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "m"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "\[ImaginaryI]", " ", "d", " ", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", "p"]], ")"]], " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "-", "p"]], ")"]], " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 z α - 1 p z cos m ( d + c z ) cosh ( a z ) z 2 - m - 1 z α ( ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( E TagBox["E", ExpIntegralE] 1 - α ( ( a - p ) z ) + E TagBox["E", ExpIntegralE] 1 - α ( - ( a + p ) z ) ) ( m mod 2 \$CellContext`m 2 - 1 ) - k = 0 m - 1 2 - d ( 2 k + m ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 d m ( E TagBox["E", ExpIntegralE] 1 - α ( ( a + 2 c k - c m - p ) z ) + E TagBox["E", ExpIntegralE] 1 - α ( - ( a - 2 c k + c m + p ) z ) ) + 4 d k ( E TagBox["E", ExpIntegralE] 1 - α ( ( a + c ( m - 2 k ) - p ) z ) + E TagBox["E", ExpIntegralE] 1 - α ( - ( a + 2 c k - c m + p ) z ) ) ) ) /; m + Condition z z α -1 p z d c z m a z 2 -1 m -1 z α Binomial m m 2 -1 ExpIntegralE 1 -1 α a -1 p z ExpIntegralE 1 -1 α -1 a p z \$CellContext`m 2 -1 -1 k 0 m -1 2 -1 -1 d 2 k m Binomial m k 2 d m ExpIntegralE 1 -1 α a 2 c k -1 c m -1 p z ExpIntegralE 1 -1 α -1 a -1 2 c k c m p z 4 d k ExpIntegralE 1 -1 α a c m -1 2 k -1 p z ExpIntegralE 1 -1 α -1 a 2 c k -1 c m p z m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["d_", "+", RowBox[List["c_", " ", "z_"]]]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "p"]], ")"]], " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "m"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "\[ImaginaryI]", " ", "d", " ", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", "p"]], ")"]], " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "-", "p"]], ")"]], " ", "z"]]]], "]"]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18