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 Tan

 http://functions.wolfram.com/01.08.21.0073.01

 Input Form

 Integrate[E^(p z) Cos[b z]^m Tan[c z], z] == ((-I) ((1/(p (2 I c + p))) (Binomial[m, m/2] (E^(p z) (2 I c + p) Hypergeometric2F1[-((I p)/(2 c)), 1, 1 - (I p)/(2 c), -E^(2 I c z)] - E^((2 I c + p) z) p Hypergeometric2F1[1 - (I p)/(2 c), 1, 2 - (I p)/(2 c), -E^(2 I c z)]) (-1 + Mod[m, 2])) + Sum[Binomial[m, k] (-(I E^((2 I c - I b (-2 k + m) + p) z) Hypergeometric2F1[(2 c + 2 b k - b m - I p)/(2 c), 1, (4 c + 2 b k - b m - I p)/(2 c), -E^(2 I c z)])/ (2 c + 2 b k - b m - I p) + (E^((I b (-2 k + m) + p) z) Hypergeometric2F1[(-2 b k + b m - I p)/(2 c), 1, (2 c - 2 b k + b m - I p)/(2 c), -E^(2 I c z)])/ (2 I b k - I b m - p) - (I E^((2 I c + I b (-2 k + m) + p) z) Hypergeometric2F1[(2 c - 2 b k + b m - I p)/(2 c), 1, (4 c - 2 b k + b m - I p)/(2 c), -E^(2 I c z)])/ (2 c - 2 b k + b m - I p) - (E^(((-I) b (-2 k + m) + p) z) Hypergeometric2F1[-((-2 b k + b m + I p)/(2 c)), 1, (2 c + 2 b k - b m - I p)/(2 c), -E^(2 I c z)])/ (2 I b k - I b m + p)), {k, 0, Floor[(1/2) (-1 + m)]}]))/2^m /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 p z cos m ( b z ) tan ( c z ) z - 2 - m ( 1 p ( 2 c + p ) ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p z ( 2 c + p ) 2 F 1 ( - p 2 c , 1 ; 1 - p 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[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( 2 c + p ) z p 2 F 1 ( 1 - p 2 c , 1 ; 2 - p 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[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "p"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ( m mod 2 \$CellContext`m 2 - 1 ) + k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b ( m - 2 k ) + p ) z 2 F 1 ( - 2 b k + b m - p 2 c , 1 ; 2 c - 2 b k + b m - p 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[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] 2 b k - b m - p - ( ( 2 c - b ( m - 2 k ) + p ) z 2 F 1 ( 2 c + 2 b k - b m - p 2 c , 1 ; 4 c + 2 b k - b m - p 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["2", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) / ( 2 c + 2 b k - b m - p ) - ( p - b ( m - 2 k ) ) z 2 F 1 ( - - 2 b k + b m + p 2 c , 1 ; 2 c + 2 b k - b m - p 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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] 2 b k - b m + p - ( ( 2 c + b ( m - 2 k ) + p ) z 2 F 1 ( 2 c - 2 b k + b m - p 2 c , 1 ; 4 c - 2 b k + b m - p 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["2", " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["4", " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) / ( 2 c - 2 b k + b m - p ) ) ) /; m + Condition z p z b z m c z -1 2 -1 m 1 p 2 c p -1 Binomial m m 2 -1 p z 2 c p Hypergeometric2F1 -1 p 2 c -1 1 1 -1 p 2 c -1 -1 2 c z -1 2 c p z p Hypergeometric2F1 1 -1 p 2 c -1 1 2 -1 p 2 c -1 -1 2 c z \$CellContext`m 2 -1 k 0 m -1 2 -1 Binomial m k b m -1 2 k p z Hypergeometric2F1 -2 b k b m -1 p 2 c -1 1 2 c -1 2 b k b m -1 p 2 c -1 -1 2 c z 2 b k -1 b m -1 p -1 -1 2 c -1 b m -1 2 k p z Hypergeometric2F1 2 c 2 b k -1 b m -1 p 2 c -1 1 4 c 2 b k -1 b m -1 p 2 c -1 -1 2 c z 2 c 2 b k -1 b m -1 p -1 -1 p -1 b m -1 2 k z Hypergeometric2F1 -1 -2 b k b m p 2 c -1 1 2 c 2 b k -1 b m -1 p 2 c -1 -1 2 c z 2 b k -1 b m p -1 -1 2 c b m -1 2 k p z Hypergeometric2F1 2 c -1 2 b k b m -1 p 2 c -1 1 4 c -1 2 b k b m -1 p 2 c -1 -1 2 c z 2 c -1 2 b k b m -1 p -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18