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 Tanh

 http://functions.wolfram.com/01.21.21.0153.01

 Input Form

 Integrate[E^(p z) Cosh[b z] Tanh[c z], z] == (1/2) ((E^((-b + p) z) Hypergeometric2F1[(-b + p)/(2 c), 1, 1 + (-b + p)/(2 c), -E^(2 c z)])/(b - p) - (E^((b + p) z) Hypergeometric2F1[(b + p)/(2 c), 1, 1 + (b + p)/(2 c), -E^(2 c z)])/(b + p) - (E^((-b + p + 2 c) z) Hypergeometric2F1[(-b + p + 2 c)/(2 c), 1, (-b + p + 4 c)/(2 c), -E^(2 c z)])/(b - p - 2 c) + (E^((b + p + 2 c) z) Hypergeometric2F1[(b + p + 2 c)/(2 c), 1, (b + p + 4 c)/(2 c), -E^(2 c z)])/(b + p + 2 c))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p"]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "-", "p"]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "+", "p"]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["2", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["4", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "-", "p", "-", RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List["b", "+", "p", "+", RowBox[List["4", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c"]]]]]]], ")"]]]]]]]]

 MathML Form

 p z cosh ( b z ) tanh ( c z ) z 1 2 ( ( p - b ) z b - p 2 F 1 ( p - b 2 c , 1 ; p - b 2 c + 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["p", "-", "b"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", "b"]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + ( b + p + 2 c ) z b + p + 2 c 2 F 1 ( b + p + 2 c 2 c , 1 ; b + p + 4 c 2 c ; - 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["b", "+", "p", "+", RowBox[List["2", " ", "c"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["b", "+", "p", "+", RowBox[List["4", " ", "c"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( b + p ) z b + p 2 F 1 ( b + p 2 c , 1 ; b + p 2 c + 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["b", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( - b + p + 2 c ) z b - p - 2 c 2 F 1 ( - b + p + 2 c 2 c , 1 ; - b + p + 4 c 2 c ; - 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[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["2", " ", "c"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["4", " ", "c"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) z p z b z c z 1 2 p -1 b z b -1 p -1 Hypergeometric2F1 p -1 b 2 c -1 1 p -1 b 2 c -1 1 -1 2 c z b p 2 c z b p 2 c -1 Hypergeometric2F1 b p 2 c 2 c -1 1 b p 4 c 2 c -1 -1 2 c z -1 b p z b p -1 Hypergeometric2F1 b p 2 c -1 1 b p 2 c -1 1 -1 2 c z -1 -1 b p 2 c z b -1 p -1 2 c -1 Hypergeometric2F1 -1 b p 2 c 2 c -1 1 -1 b p 4 c 2 c -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_"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p"]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "-", "p"]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", RowBox[List["1", "+", FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "+", "p"]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["2", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p", "+", RowBox[List["4", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "-", "p", "-", RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List["b", "+", "p", "+", RowBox[List["4", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List["b", "+", "p", "+", RowBox[List["2", " ", "c"]]]]]]], ")"]]]]]]]]

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

 2002-12-18