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 Cosh

 http://functions.wolfram.com/01.20.21.1285.01

 Input Form

 Integrate[E^(p z) Cos[c z]^\[Mu] Cosh[b + a z], z] == ((1/2) Cos[c z]^\[Mu] ((-(1/(a - p + I c \[Mu]))) (E^((-a + p) z) Hypergeometric2F1[(I (a - p + I c \[Mu]))/(2 c), -\[Mu], 1 + (I (a - p))/(2 c) - \[Mu]/2, -E^(2 I c z)]) + (1/(a + p - I c \[Mu])) (E^(2 b + (a + p) z) Hypergeometric2F1[ -((I (a + p) + c \[Mu])/(2 c)), -\[Mu], 1 - (I (a + p))/(2 c) - \[Mu]/2, -E^(2 I c z)])))/(E^b (1 + E^(2 I c z))^\[Mu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["a", "-", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["a", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 p z cos μ ( c z ) cosh ( b + a z ) z 1 2 - b ( 1 + 2 c z ) - μ cos μ ( c z ) ( 2 b + ( a + p ) z a + p - c μ 2 F 1 ( - ( a + p ) + c μ 2 c , - μ ; - ( a + p ) 2 c - μ 2 + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], "-", FractionBox["\[Mu]", "2"], "+", "1"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( p - a ) z a - p + c μ 2 F 1 ( ( a - p + c μ ) 2 c , - μ ; ( a - p ) 2 c - μ 2 + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"], "+", "1"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) z p z c z μ b a z 1 2 -1 b 1 2 c z -1 μ c z μ 2 b a p z a p -1 c μ -1 Hypergeometric2F1 -1 a p c μ 2 c -1 -1 μ -1 a p 2 c -1 -1 μ 2 -1 1 -1 2 c z -1 p -1 a z a -1 p c μ -1 Hypergeometric2F1 a -1 p c μ 2 c -1 -1 μ a -1 p 2 c -1 -1 μ 2 -1 1 -1 2 c z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]], "\[Mu]_"], " ", RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]], ")"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "-", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["a", "-", "p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["a", "+", "p"]], ")"]]]], RowBox[List["2", " ", "c"]]], "-", FractionBox["\[Mu]", "2"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["a", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "\[Mu]"]]]]]]], ")"]]]]]]]]

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

 2002-12-18