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 Cosh

 http://functions.wolfram.com/01.20.21.3782.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18