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 Cosh

 http://functions.wolfram.com/01.20.21.4180.01

 Input Form

 Integrate[Sinh[d z]^m Cosh[c Sqrt[z] + g]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) + (2^(-m - v)/d) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k E^(d (2 k - m) z) (-(-1)^m + E^(2 (-2 k + m) d z)) Binomial[m, k])/(-2 k + m), {k, 0, Floor[(1/2) (-1 + m)]}] + (1/c^2) (I^m 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(-2 s + v)^2) (E^((-2 s - v) (g + c Sqrt[z])) (E^(4 s (g + c Sqrt[z])) (-1 + c (2 s - v) Sqrt[z]) + E^(2 v (g + c Sqrt[z])) (-1 + c (-2 s + v) Sqrt[z])) Binomial[v, s]), {s, 0, Floor[(1/2) (-1 + v)]}]) + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (((2 (-1)^m E^(2 g (-2 s + v) + c (-2 s + v) Sqrt[z] + d (2 k - m) z))/(d (2 k - m)) + (2 E^((-c) (-2 s + v) Sqrt[z] + d (-2 k + m) z))/(d (-2 k + m)) - (1/(d (2 k - m))^(3/2)) ((-1)^m c E^(-((c^2 (-2 s + v)^2)/ (4 d (2 k - m))) + 2 g (-2 s + v)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 d (2 k - m) Sqrt[z])/(2 Sqrt[d (2 k - m)])]) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 d (2 k - m) Sqrt[z])/(2 Sqrt[d (-2 k + m)])])/(E^((c^2 (-2 s + v)^2)/(4 d (-2 k + m))) (d (-2 k + m))^(3/2)))/E^(g (-2 s + v)) + ((-1)^m ((2 E^((-c) (-2 s + v) Sqrt[z] + d (2 k - m) z))/ (d (2 k - m)) + (2 (-1)^m E^(2 g (-2 s + v) + c (-2 s + v) Sqrt[z] + d (-2 k + m) z))/(d (-2 k + m)) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 d (-2 k + m) Sqrt[z])/(2 Sqrt[d (2 k - m)])])/(E^((c^2 (-2 s + v)^2)/ (4 d (2 k - m))) (d (2 k - m))^(3/2)) - (1/(d (-2 k + m))^(3/2)) ((-1)^m c E^(-((c^2 (-2 s + v)^2)/ (4 d (-2 k + m))) + 2 g (-2 s + v)) Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 d (-2 k + m) Sqrt[z])/ (2 Sqrt[d (-2 k + m)])])))/E^(g (-2 s + v))), {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

 sinh m ( d z ) cosh v ( z c + g ) z m 2 - m - v 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) + 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]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) d k = 0 m - 1 2 ( - 1 ) k d ( 2 k - m ) z ( - ( - 1 ) m + 2 ( m - 2 k ) d z ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] m - 2 k + m 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 ) c 2 s = 0 v - 1 2 1 ( v - 2 s ) 2 ( ( - 2 s - v ) ( z c + g ) ( 4 s ( z c + g ) ( c ( 2 s - v ) z - 1 ) + 2 v ( z c + g ) ( c ( v - 2 s ) z - 1 ) ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) + 2 - m - v - 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]]]]] ( - g ( v - 2 s ) ( - 1 ( d ( 2 k - m ) ) 3 / 2 ( ( - 1 ) m c 2 g ( v - 2 s ) - c 2 ( v - 2 s ) 2 4 d ( 2 k - m ) π ( v - 2 s ) erfi ( 2 d z ( 2 k - m ) + c ( v - 2 s ) 2 d ( 2 k - m ) ) ) + 2 ( - 1 ) m 2 g ( v - 2 s ) + c z ( v - 2 s ) + d ( 2 k - m ) z d ( 2 k - m ) + 2 d ( m - 2 k ) z - c ( v - 2 s ) z d ( m - 2 k ) - c - c 2 ( v - 2 s ) 2 4 d ( m - 2 k ) π ( v - 2 s ) erfi ( 2 d z ( 2 k - m ) + c ( v - 2 s ) 2 d ( m - 2 k ) ) ( d ( m - 2 k ) ) 3 / 2 ) + ( - 1 ) m - g ( v - 2 s ) ( - c - c 2 ( v - 2 s ) 2 4 d ( 2 k - m ) π ( v - 2 s ) erfi ( 2 d z ( m - 2 k ) + c ( v - 2 s ) 2 d ( 2 k - m ) ) ( d ( 2 k - m ) ) 3 / 2 + 2 d ( 2 k - m ) z - c ( v - 2 s ) z d ( 2 k - m ) + 2 ( - 1 ) m 2 g ( v - 2 s ) + c z ( v - 2 s ) + d ( m - 2 k ) z d ( m - 2 k ) - ( - 1 ) m c 2 g ( v - 2 s ) - c 2 ( v - 2 s ) 2 4 d ( m - 2 k ) π ( v - 2 s ) erfi ( 2 d z ( m - 2 k ) + c ( v - 2 s ) 2 d ( m - 2 k ) ) ( d ( m - 2 k ) ) 3 / 2 ) ) /; m + v + Condition z d z m z 1 2 c g v m 2 -1 m -1 v z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 2 -1 m -1 v Binomial v v 2 -1 1 -1 \$CellContext`v 2 d -1 k 0 m -1 2 -1 -1 k d 2 k -1 m z -1 -1 m 2 m -1 2 k d z Binomial m k m -1 2 k -1 m 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c 2 -1 s 0 v -1 2 -1 1 v -1 2 s 2 -1 -2 s -1 v z 1 2 c g 4 s z 1 2 c g c 2 s -1 v z 1 2 -1 2 v z 1 2 c g c v -1 2 s z 1 2 -1 Binomial v s 2 -1 m -1 v -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 g v -1 2 s -1 1 d 2 k -1 m 3 2 -1 -1 m c 2 g v -1 2 s -1 c 2 v -1 2 s 2 4 d 2 k -1 m -1 1 2 v -1 2 s Erfi 2 d z 1 2 2 k -1 m c v -1 2 s 2 d 2 k -1 m 1 2 -1 2 -1 m 2 g v -1 2 s c z 1 2 v -1 2 s d 2 k -1 m z d 2 k -1 m -1 2 d m -1 2 k z -1 c v -1 2 s z 1 2 d m -1 2 k -1 -1 c -1 c 2 v -1 2 s 2 4 d m -1 2 k -1 1 2 v -1 2 s Erfi 2 d z 1 2 2 k -1 m c v -1 2 s 2 d m -1 2 k 1 2 -1 d m -1 2 k 3 2 -1 -1 m -1 g v -1 2 s -1 c -1 c 2 v -1 2 s 2 4 d 2 k -1 m -1 1 2 v -1 2 s Erfi 2 d z 1 2 m -1 2 k c v -1 2 s 2 d 2 k -1 m 1 2 -1 d 2 k -1 m 3 2 -1 2 d 2 k -1 m z -1 c v -1 2 s z 1 2 d 2 k -1 m -1 2 -1 m 2 g v -1 2 s c z 1 2 v -1 2 s d m -1 2 k z d m -1 2 k -1 -1 -1 m c 2 g v -1 2 s -1 c 2 v -1 2 s 2 4 d m -1 2 k -1 1 2 v -1 2 s Erfi 2 d z 1 2 m -1 2 k c v -1 2 s 2 d m -1 2 k 1 2 -1 d m -1 2 k 3 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18