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 Cosh

 http://functions.wolfram.com/01.20.21.0275.01

 Input Form

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

 Standard Form

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

 MathML Form

 p z ( a + b d z ) β cosh ( e + c z ) z 1 2 ( c - p ) ( c + p ) ( ( a + b d z ) β ( d z b a + 1 ) - β ( e + ( c + p ) z ( c - p ) 2 F 1 ( c + p d , - β ; c + d + p 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", "+", "p"]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "+", "p"]], "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] - ( p - c ) z - e ( c + p ) 2 F 1 ( p - c d , - β ; - c + d + p 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["p", "-", "c"]], "d"], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Beta]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", "p"]], "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 p z a b d z β e c z 1 2 c -1 p c p -1 a b d z β d z b a -1 1 -1 β e c p z c -1 p Hypergeometric2F1 c p d -1 -1 β c d p d -1 -1 b d z a -1 -1 p -1 c z -1 e c p Hypergeometric2F1 p -1 c d -1 -1 β -1 c d p d -1 -1 b d z a -1 [/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["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d_", " ", "z_"]]]]]]], ")"]], "\[Beta]_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", "z_"]], "+", "e_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]]]], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]], "-", "e"]]]]], " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "p"]], "d"], ",", RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "p"]], "d"], ",", RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["c", "+", "d", "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]]]]]]]

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

 2002-12-18