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 Cos

 http://functions.wolfram.com/01.07.21.2543.01

 Input Form

 Integrate[z^n Sin[b z^2 + e]^m Cos[c z^2 + g]^v, z] == (1/(1 + n)) (2^(-m - v) z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) - 2^(-1 - m - v) z^(1 + n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (E^(I e (-2 k + m) - (I m Pi)/2) ((-I) b (-2 k + m) z^2)^ ((1/2) (-1 - n)) Gamma[(1 + n)/2, (-I) b (-2 k + m) z^2] + E^((-I) e (-2 k + m) + (I m Pi)/2) (I b (-2 k + m) z^2)^ ((1/2) (-1 - n)) Gamma[(1 + n)/2, I b (-2 k + m) z^2]), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(-1 - m - v) z^(1 + n) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (E^(I g (-2 s + v)) ((-I) c (-2 s + v) z^2)^ ((1/2) (-1 - n)) Gamma[(1 + n)/2, (-I) c (-2 s + v) z^2] + ((I c (-2 s + v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, I c (-2 s + v) z^2])/E^(I g (-2 s + v))), {s, 0, Floor[(1/2) (-1 + v)]}] - 2^(-1 - m - v) z^(1 + n) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (E^(I e (-2 k + m) - (I m Pi)/2 + I g (-2 s + v)) (((-I) b (-2 k + m) - I c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, ((-I) b (-2 k + m) - I c (-2 s + v)) z^2] + E^((-I) e (-2 k + m) + (I m Pi)/2 + I g (-2 s + v)) ((I b (-2 k + m) - I c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (I b (-2 k + m) - I c (-2 s + v)) z^2] + E^(I e (-2 k + m) - (I m Pi)/2 - I g (-2 s + v)) (((-I) b (-2 k + m) + I c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, ((-I) b (-2 k + m) + I c (-2 s + v)) z^2] + E^((-I) e (-2 k + m) + (I m Pi)/2 - I g (-2 s + v)) ((I b (-2 k + m) + I c (-2 s + v)) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (I b (-2 k + m) + I c (-2 s + v)) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin m ( b z 2 + e ) cos v ( c z 2 + g ) z - 2 - m - v - 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) z n + 1 k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( e ( m - 2 k ) - m π 2 Γ ( n + 1 2 , - b ( m - 2 k ) z 2 ) ( - b ( m - 2 k ) z 2 ) 1 2 ( - n - 1 ) + m π 2 - e ( m - 2 k ) ( b ( m - 2 k ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , b ( m - 2 k ) z 2 ) ) - 2 - m - v - 1 ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) z n + 1 s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( g ( v - 2 s ) Γ ( n + 1 2 , - c ( v - 2 s ) z 2 ) ( - c ( v - 2 s ) z 2 ) 1 2 ( - n - 1 ) + - g ( v - 2 s ) ( c ( v - 2 s ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , c ( v - 2 s ) z 2 ) ) - 2 - m - v - 1 z n + 1 k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 1 2 π m + e ( m - 2 k ) - g ( v - 2 s ) Γ ( n + 1 2 , ( c ( v - 2 s ) - b ( m - 2 k ) ) z 2 ) ( ( c ( v - 2 s ) - b ( m - 2 k ) ) z 2 ) 1 2 ( - n - 1 ) + π m 2 - e ( m - 2 k ) - g ( v - 2 s ) ( ( b ( m - 2 k ) + c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( b ( m - 2 k ) + c ( v - 2 s ) ) z 2 ) + - 1 2 π m + e ( m - 2 k ) + g ( v - 2 s ) ( ( - b ( m - 2 k ) - c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( - b ( m - 2 k ) - c ( v - 2 s ) ) z 2 ) + π m 2 - e ( m - 2 k ) + g ( v - 2 s ) ( ( b ( m - 2 k ) - c ( v - 2 s ) ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , ( b ( m - 2 k ) - c ( v - 2 s ) ) z 2 ) ) + 2 - m - v z n + 1 ( 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]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) n + 1 /; n m + v + Condition z z n b z 2 e m c z 2 g v -1 2 -1 m -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 z n 1 k 0 m -1 2 -1 -1 k Binomial m k e m -1 2 k -1 m 2 -1 Gamma n 1 2 -1 -1 b m -1 2 k z 2 -1 b m -1 2 k z 2 1 2 -1 n -1 m 2 -1 -1 e m -1 2 k b m -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 b m -1 2 k z 2 -1 2 -1 m -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 z n 1 s 0 v -1 2 -1 Binomial v s g v -1 2 s Gamma n 1 2 -1 -1 c v -1 2 s z 2 -1 c v -1 2 s z 2 1 2 -1 n -1 -1 g v -1 2 s c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 c v -1 2 s z 2 -1 2 -1 m -1 v -1 z n 1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 1 2 m e m -1 2 k -1 g v -1 2 s Gamma n 1 2 -1 c v -1 2 s -1 b m -1 2 k z 2 c v -1 2 s -1 b m -1 2 k z 2 1 2 -1 n -1 m 2 -1 -1 e m -1 2 k -1 g v -1 2 s b m -1 2 k c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 b m -1 2 k c v -1 2 s z 2 -1 1 2 m e m -1 2 k g v -1 2 s -1 b m -1 2 k -1 c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b m -1 2 k -1 c v -1 2 s z 2 m 2 -1 -1 e m -1 2 k g v -1 2 s b m -1 2 k -1 c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 b m -1 2 k -1 c v -1 2 s z 2 2 -1 m -1 v z n 1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 n 1 -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18