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 Cosh

 http://functions.wolfram.com/01.20.21.1108.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Cos[c z]^m Cosh[b + a z], z] == ((-2^(-1 - m)) z^\[Alpha] Binomial[m, m/2] (E^(2 b) ExpIntegralE[1 - \[Alpha], (-a) z] + ExpIntegralE[1 - \[Alpha], a z]) (1 - Mod[m, 2]))/E^b - (2^(-1 - m) z^\[Alpha] Sum[Binomial[m, s] (ExpIntegralE[1 - \[Alpha], (a + I c (m - 2 s)) z] + E^(2 b) ExpIntegralE[1 - \[Alpha], (-a + I c m - 2 I c s) z] + E^(2 b) ExpIntegralE[1 - \[Alpha], (-a - I c (m - 2 s)) z] + ExpIntegralE[1 - \[Alpha], (a - I c m + 2 I c s) z]), {s, 0, Floor[(1/2) (-1 + m)]}])/E^b /; 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[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]]], "]"]]]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "s"]], "]"]], " ", 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["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "s"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "s"]]]], ")"]], " ", "z"]]]], "]"]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18