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 Cosh

 http://functions.wolfram.com/01.20.21.1284.01

 Input Form

 Integrate[E^(p z) Cos[d + c z]^m Cosh[a z], z] == (Binomial[m, m/2] (1 - Mod[m, 2]) ((E^(p z) (p Cosh[a z] - a Sinh[a z]))/ (p^2 - a^2)))/2^m + 2^(1 - m) Sum[Binomial[m, k] ((E^(p z) (Cosh[a z] (p (-a^2 + c^2 (-2 k + m)^2 + p^2) Cos[(2 k - m) (d + c z)] + c (2 k - m) (a^2 + c^2 (-2 k + m)^2 + p^2) Sin[(2 k - m) (d + c z)]) + a ((a^2 + c^2 (-2 k + m)^2 - p^2) Cos[(2 k - m) (d + c z)] + 2 c (-2 k + m) p Sin[(2 k - m) (d + c z)]) Sinh[a z]))/ (a^4 + 2 a^2 (c^2 (-2 k + m)^2 - p^2) + (c^2 (-2 k + m)^2 + p^2)^2)), {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["\[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[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List["a", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["p", "2"], "-", SuperscriptBox["a", "2"]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", 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[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["p", " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], "2"]]], "+", SuperscriptBox["p", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], "2"]]], "+", SuperscriptBox["p", "2"]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], "2"]]], "-", SuperscriptBox["p", "2"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "p", " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], "2"]]], "-", SuperscriptBox["p", "2"]]], ")"]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], "2"]]], "+", SuperscriptBox["p", "2"]]], ")"]], "2"]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18