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 Cosh

 http://functions.wolfram.com/01.20.21.4806.01

 Input Form

 Integrate[z^(\[Alpha] - 1) E^(p z) Sinh[d + c z] Cosh[b + a z]^v, z] == 2^(-1 - v) z^\[Alpha] ((Binomial[v, v/2] (-ExpIntegralE[1 - \[Alpha], (c - p) z] + E^(2 d) ExpIntegralE[1 - \[Alpha], (-(c + p)) z]) (-1 + Mod[v, 2]))/ E^d - Sum[E^(-d - 2 b s - b v) Binomial[v, s] ((-E^(2 b v)) ExpIntegralE[1 - \[Alpha], (c - p + 2 a s - a v) z] + E^(2 d + 4 b s) ExpIntegralE[1 - \[Alpha], (-(c + p + 2 a s - a v)) z] - E^(4 b s) ExpIntegralE[1 - \[Alpha], (c - p - 2 a s + a v) z] + E^(2 (d + b v)) ExpIntegralE[1 - \[Alpha], (-(c + p - 2 a s + a v)) z]), {s, 0, Floor[(1/2) (-1 + v)]}]) /; Element[v, Integers] && v > 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"]]], " ", RowBox[List["Sinh", "[", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "d"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "d"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "v"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", "v"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "d"]], "+", RowBox[List["4", " ", "b", " ", "s"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]]]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "b", " ", "s"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["b", " ", "v"]]]], ")"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 z α - 1 p z sinh ( d + c z ) cosh v ( b + a z ) z 2 - v - 1 z α ( - d ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 d E TagBox["E", ExpIntegralE] 1 - α ( - ( c + p ) z ) - E TagBox["E", ExpIntegralE] 1 - α ( ( c - p ) z ) ) ( v mod 2 \$CellContext`v 2 - 1 ) - s = 0 v - 1 2 - d - 2 b s - b v ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 4 b s E TagBox["E", ExpIntegralE] 1 - α ( ( c - p - 2 a s + a v ) z ) + 2 ( d + b v ) E TagBox["E", ExpIntegralE] 1 - α ( - ( c + p - 2 a s + a v ) z ) - 2 b v E TagBox["E", ExpIntegralE] 1 - α ( ( c - p + 2 a s - a v ) z ) + 2 d + 4 b s E TagBox["E", ExpIntegralE] 1 - α ( - ( c + p + 2 a s - a v ) z ) ) ) /; v + Condition z z α -1 p z d c z b a z v 2 -1 v -1 z α -1 d Binomial v v 2 -1 2 d ExpIntegralE 1 -1 α -1 c p z -1 ExpIntegralE 1 -1 α c -1 p z \$CellContext`v 2 -1 -1 s 0 v -1 2 -1 -1 d -1 2 b s -1 b v Binomial v s -1 4 b s ExpIntegralE 1 -1 α c -1 p -1 2 a s a v z 2 d b v ExpIntegralE 1 -1 α -1 c p -1 2 a s a v z -1 2 b v ExpIntegralE 1 -1 α c -1 p 2 a s -1 a v z 2 d 4 b s ExpIntegralE 1 -1 α -1 c p 2 a s -1 a v z v 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_"]]], " ", RowBox[List["Sinh", "[", RowBox[List["d_", "+", RowBox[List["c_", " ", "z_"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "d"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "d"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "v"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b", " ", "v"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "d"]], "+", RowBox[List["4", " ", "b", " ", "s"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]]]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "b", " ", "s"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["b", " ", "v"]]]], ")"]]]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", "p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18