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 Cosh

 http://functions.wolfram.com/01.20.21.1205.01

 Input Form

 Integrate[(a + b E^(d z))^\[Beta] Sin[e z] Cosh[c z], z] == ((1/4) I (a + b E^(d z))^\[Beta] ((1/(c - I e)) (E^((-c) z + I e z) (Hypergeometric2F1[-((c - I e)/d), -\[Beta], (-c + d + I e)/d, -((b E^(d z))/a)] + E^(2 (c - I e) z) Hypergeometric2F1[(c - I e)/d, -\[Beta], (c + d - I e)/d, -((b E^(d z))/a)])) - (1/(c + I e)) ((Hypergeometric2F1[-((c + I e)/d), -\[Beta], (-c + d - I e)/d, -((b E^(d z))/a)] + E^(2 (c + I e) z) Hypergeometric2F1[(c + I e)/d, -\[Beta], (c + d + I e)/d, -((b E^(d z))/a)])/E^((c + I e) z))))/(1 + (b E^(d z))/a)^\[Beta]

 Standard Form

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 MathML Form

 ( a + b d z ) β sin ( e z ) cosh ( c z ) z 1 4 ( a + b d z ) β ( d z b a + 1 ) - β ( 1 c - e ( e z - c z ( 2 ( c - e ) z 2 F 1 ( c - e d , - β ; c + d - e d ; - b d z a ) 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["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 F 1 ( - c - e d , - β ; - c + d + e d ; - b d z a ) 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["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) - 1 c + e ( - ( c + e ) z ( 2 ( c + e ) z 2 F 1 ( c + e d , - β ; c + d + e d ; - b d z a ) 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["c", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 F 1 ( - c + e d , - β ; - c + d - e d ; - b d z a ) 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["c", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) ) z a b d z β e z c z 1 4 a b d z β d z b a -1 1 -1 β 1 c -1 e -1 e z -1 c z 2 c -1 e z Hypergeometric2F1 c -1 e d -1 -1 β c d -1 e d -1 -1 b d z a -1 Hypergeometric2F1 -1 c -1 e d -1 -1 β -1 c d e d -1 -1 b d z a -1 -1 1 c e -1 -1 c e z 2 c e z Hypergeometric2F1 c e d -1 -1 β c d e d -1 -1 b d z a -1 Hypergeometric2F1 -1 c e d -1 -1 β -1 c d -1 e d -1 -1 b d z a -1 [/itex]

 Rule Form

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

 2002-12-18