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 Tan

 http://functions.wolfram.com/01.08.21.0046.01

 Input Form

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

 Standard Form

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 MathML Form

 cos m ( b z ) tan ( c z ) z - 2 - m log ( cos ( c z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) c ( 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]]]]] + 2 - m 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 ) z 2 F 1 ( b ( m - 2 k ) 2 c , 1 ; 2 c + b ( m - 2 k ) 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", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], 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] b ( m - 2 k ) - - b ( m - 2 k ) z 2 F 1 ( - b ( m - 2 k ) 2 c , 1 ; 2 c - b ( m - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], 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] b ( m - 2 k ) - ( 2 c + b ( m - 2 k ) ) z 2 F 1 ( 2 c + b ( m - 2 k ) 2 c , 1 ; 4 c + b ( m - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], 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 + b ( m - 2 k ) - ( 2 c - b ( m - 2 k ) ) z 2 F 1 ( 2 c - b ( m - 2 k ) 2 c , 1 ; 4 c - b ( m - 2 k ) 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], 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["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], 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 - b ( m - 2 k ) ) /; m + Condition z b z m c z -1 2 -1 m c z 1 -1 \$CellContext`m 2 c -1 Binomial m m 2 -1 2 -1 m k 0 m -1 2 -1 Binomial m k b m -1 2 k z Hypergeometric2F1 b m -1 2 k 2 c -1 1 2 c b m -1 2 k 2 c -1 -1 2 c z b m -1 2 k -1 -1 -1 b m -1 2 k z Hypergeometric2F1 -1 b m -1 2 k 2 c -1 1 2 c -1 b m -1 2 k 2 c -1 -1 2 c z b m -1 2 k -1 -1 2 c b m -1 2 k z Hypergeometric2F1 2 c b m -1 2 k 2 c -1 1 4 c b m -1 2 k 2 c -1 -1 2 c z 2 c b m -1 2 k -1 -1 2 c -1 b m -1 2 k z Hypergeometric2F1 2 c -1 b m -1 2 k 2 c -1 1 4 c -1 b m -1 2 k 2 c -1 -1 2 c z 2 c -1 b m -1 2 k -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18