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 Cosh

 http://functions.wolfram.com/01.20.21.3837.01

 Input Form

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

 Standard Form

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 MathML Form

 p z cos m ( d + c z ) cosh v ( b + a z ) z 2 - m - v ( ( 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]]]]] ( s = 0 m - 1 2 - d ( m - 2 s ) ( 2 d ( m - 2 s ) - ( - p - c ( m - 2 s ) ) z - p - c ( m - 2 s ) + - ( c ( m - 2 s ) - p ) z c ( m - 2 s ) - p ) ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) ( v mod 2 \$CellContext`v 2 - 1 ) + p z ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) s = 0 v - 1 2 - b ( 2 s + v ) ( 2 b v - ( - p - a ( v - 2 s ) ) z - p - a ( v - 2 s ) + 4 b s - ( - p - 2 a s + a v ) z - p - 2 a s + a v ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] - 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]]]]] s = 0 v - 1 2 - d ( 2 k + m ) - b ( 2 s + v ) ( 2 d m ( 2 b v - ( c ( 2 k - m ) - p + 2 a s - a v ) z c ( 2 k - m ) - p + 2 a s - a v + 4 b s - ( c ( 2 k - m ) - p - 2 a s + a v ) z c ( 2 k - m ) - p - 2 a s + a v ) + 4 d k ( 2 b v - ( - 2 c k + c m - p + 2 a s - a v ) z - 2 c k + c m - p + 2 a s - a v + 4 b s - ( c ( m - 2 k ) - p - 2 a s + a v ) z c ( m - 2 k ) - p - 2 a s + a v ) ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z d c z m b a z v 2 -1 m -1 v Binomial v v 2 -1 s 0 m -1 2 -1 -1 d m -1 2 s 2 d m -1 2 s -1 -1 p -1 c m -1 2 s z -1 p -1 c m -1 2 s -1 -1 c m -1 2 s -1 p z c m -1 2 s -1 p -1 Binomial m s \$CellContext`v 2 -1 p z Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 p -1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 -1 b 2 s v 2 b v -1 -1 p -1 a v -1 2 s z -1 p -1 a v -1 2 s -1 4 b s -1 -1 p -1 2 a s a v z -1 p -1 2 a s a v -1 Binomial v s -1 k 0 m -1 2 -1 Binomial m k s 0 v -1 2 -1 -1 d 2 k m -1 b 2 s v 2 d m 2 b v -1 c 2 k -1 m -1 p 2 a s -1 a v z c 2 k -1 m -1 p 2 a s -1 a v -1 4 b s -1 c 2 k -1 m -1 p -1 2 a s a v z c 2 k -1 m -1 p -1 2 a s a v -1 4 d k 2 b v -1 -2 c k c m -1 p 2 a s -1 a v z -2 c k c m -1 p 2 a s -1 a v -1 4 b s -1 c m -1 2 k -1 p -1 2 a s a v z c m -1 2 k -1 p -1 2 a s a v -1 Binomial v s m m 0 v v 0 [/itex]

 Rule Form

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

 2002-12-18