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 Cosh

 http://functions.wolfram.com/01.20.21.0608.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18