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 Cosh

 http://functions.wolfram.com/01.20.21.1317.01

 Input Form

 Integrate[(E^(p z) Cosh[c z])/(a + b Cosh[c z]^2), z] == (1/(2 Sqrt[a] b Sqrt[a + b])) (-((1/(c + p)) (E^((c + p) z) ((-(2 a + b + 2 Sqrt[a] Sqrt[a + b])) Hypergeometric2F1[(c + p)/(2 c), 1, (3 c + p)/(2 c), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[(c + p)/(2 c), 1, (3 c + p)/(2 c), -((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))) - (1/(3 c + p)) (E^((3 c + p) z) ((-(2 a + b + 2 Sqrt[a] Sqrt[a + b])) Hypergeometric2F1[(3 c + p)/(2 c), 1, (5 c + p)/(2 c), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[(3 c + p)/(2 c), 1, (5 c + p)/(2 c), -((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox["a"], " ", "b", " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["c", "+", "p"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["5", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["5", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 p z cosh ( c z ) a + b cosh 2 ( c z ) z 1 2 a b a + b ( - 1 c + p ( ( c + p ) z ( ( 2 a - 2 a + b a + b ) 2 F 1 ( c + p 2 c , 1 ; 3 c + p 2 c ; - b 2 c z 2 a + 2 a + b a + b ) 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"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", SqrtBox["a"]]], "+", "b"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( 2 a + 2 a + b a + b ) 2 F 1 ( c + p 2 c , 1 ; 3 c + p 2 c ; - b 2 c z 2 a - 2 a + b a + b ) 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"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", SqrtBox["a"]]], "+", "b"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) - 1 3 c + p ( ( 3 c + p ) z ( ( 2 a - 2 a + b a + b ) 2 F 1 ( 3 c + p 2 c , 1 ; 5 c + p 2 c ; - b 2 c z 2 a + 2 a + b a + b ) 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[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["5", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", SqrtBox["a"]]], "+", "b"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( 2 a + 2 a + b a + b ) 2 F 1 ( 3 c + p 2 c , 1 ; 5 c + p 2 c ; - b 2 c z 2 a - 2 a + b a + b ) 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[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["5", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", SqrtBox["a"]]], "+", "b"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) ) z p z c z a b c z 2 -1 1 2 a 1 2 b a b 1 2 -1 -1 1 c p -1 c p z 2 a -1 2 a b 1 2 a 1 2 b Hypergeometric2F1 c p 2 c -1 1 3 c p 2 c -1 -1 b 2 c z 2 a 2 a b 1 2 a 1 2 b -1 -1 2 a 2 a b 1 2 a 1 2 b Hypergeometric2F1 c p 2 c -1 1 3 c p 2 c -1 -1 b 2 c z 2 a -1 2 a b 1 2 a 1 2 b -1 -1 1 3 c p -1 3 c p z 2 a -1 2 a b 1 2 a 1 2 b Hypergeometric2F1 3 c p 2 c -1 1 5 c p 2 c -1 -1 b 2 c z 2 a 2 a b 1 2 a 1 2 b -1 -1 2 a 2 a b 1 2 a 1 2 b Hypergeometric2F1 3 c p 2 c -1 1 5 c p 2 c -1 -1 b 2 c z 2 a -1 2 a b 1 2 a 1 2 b -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["c", "+", "p"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["5", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["5", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["3", " ", "c"]], "+", "p"]]]]], RowBox[List["2", " ", SqrtBox["a"], " ", "b", " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]

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

 2002-12-18