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 Cosh

 http://functions.wolfram.com/01.20.21.4158.01

 Input Form

 Integrate[Sinh[b Sqrt[z]]^m Cosh[f 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]) + (1/b^2) (2^(-1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/(-2 k + m)^2) (((-1)^k (E^(I m Pi + 4 b k Sqrt[z]) (-4 + 4 b (2 k - m) Sqrt[z]) + 4 E^(2 m b Sqrt[z]) (-1 + b (-2 k + m) Sqrt[z])) Binomial[m, k])/ E^(b (2 k + m) Sqrt[z])), {k, 0, Floor[(1/2) (-1 + m)]}]) + I^m 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Sinh[(2 s - v) (g + f z)] Binomial[v, s])/(f (2 s - v)), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((-((2 E^((-b) (2 k - m) Sqrt[z] - f (-2 s + v) z))/(f (-2 s + v))) + (2 (-1)^m E^(2 g (-2 s + v) + b (2 k - m) Sqrt[z] + f (-2 s + v) z))/(f (-2 s + v)) - (b E^((b^2 (2 k - m)^2)/(4 f (-2 s + v))) (2 k - m) Sqrt[Pi] Erfi[(b (2 k - m) + 2 f (-2 s + v) Sqrt[z])/(2 Sqrt[(-f) (-2 s + v)])])/ ((-f) (-2 s + v))^(3/2) - (1/(f (-2 s + v))^(3/2)) ((-1)^m b E^(-((b^2 (2 k - m)^2)/(4 f (-2 s + v))) + 2 g (-2 s + v)) (2 k - m) Sqrt[Pi] Erfi[(b (2 k - m) + 2 f (-2 s + v) Sqrt[z])/(2 Sqrt[f (-2 s + v)])]))/ E^(g (-2 s + v)) + ((-1)^m (-((2 E^((-b) (-2 k + m) Sqrt[z] - f (-2 s + v) z))/ (f (-2 s + v))) + (2 (-1)^m E^(2 g (-2 s + v) + b (-2 k + m) Sqrt[z] + f (-2 s + v) z))/(f (-2 s + v)) - (b E^((b^2 (-2 k + m)^2)/(4 f (-2 s + v))) (-2 k + m) Sqrt[Pi] Erfi[(b (-2 k + m) + 2 f (-2 s + v) Sqrt[z])/ (2 Sqrt[(-f) (-2 s + v)])])/((-f) (-2 s + v))^(3/2) - (1/(f (-2 s + v))^(3/2)) ((-1)^m b E^(-((b^2 (-2 k + m)^2)/ (4 f (-2 s + v))) + 2 g (-2 s + v)) (-2 k + m) Sqrt[Pi] Erfi[(b (-2 k + m) + 2 f (-2 s + v) Sqrt[z])/ (2 Sqrt[f (-2 s + v)])])))/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 ( b z ) cosh v ( g + f z ) 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 - 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 ) b 2 k = 0 m - 1 2 1 ( m - 2 k ) 2 ( ( - 1 ) k - b ( 2 k + m ) z ( 4 b z k + m π ( 4 b ( 2 k - m ) z - 4 ) + 4 2 m b z ( b ( m - 2 k ) z - 1 ) ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) + 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 ) s = 0 v - 1 2 sinh ( ( 2 s - v ) ( g + f z ) ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] f ( 2 s - v ) + 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 ) ( - b b 2 ( 2 k - m ) 2 4 f ( v - 2 s ) ( 2 k - m ) π erfi ( b ( 2 k - m ) + 2 f ( v - 2 s ) z 2 - f ( v - 2 s ) ) ( - f ( v - 2 s ) ) 3 / 2 + 2 ( - 1 ) m b z ( 2 k - m ) + 2 g ( v - 2 s ) + f ( v - 2 s ) z f ( v - 2 s ) - 2 - b z ( 2 k - m ) - f ( v - 2 s ) z f ( v - 2 s ) - 1 ( f ( v - 2 s ) ) 3 / 2 ( ( - 1 ) m b 2 g ( v - 2 s ) - b 2 ( 2 k - m ) 2 4 f ( v - 2 s ) ( 2 k - m ) π erfi ( b ( 2 k - m ) + 2 f ( v - 2 s ) z 2 f ( v - 2 s ) ) ) ) + ( - 1 ) m - g ( v - 2 s ) ( - b b 2 ( m - 2 k ) 2 4 f ( v - 2 s ) ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 f ( v - 2 s ) z 2 - f ( v - 2 s ) ) ( - f ( v - 2 s ) ) 3 / 2 + 2 ( - 1 ) m b z ( m - 2 k ) + 2 g ( v - 2 s ) + f ( v - 2 s ) z f ( v - 2 s ) - 2 - b z ( m - 2 k ) - f ( v - 2 s ) z f ( v - 2 s ) - 1 ( f ( v - 2 s ) ) 3 / 2 ( ( - 1 ) m b 2 g ( v - 2 s ) - b 2 ( m - 2 k ) 2 4 f ( v - 2 s ) ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 f ( v - 2 s ) z 2 f ( v - 2 s ) ) ) ) ) /; m + v + Condition z b z 1 2 m g f z 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 -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 b 2 -1 k 0 m -1 2 -1 1 m -1 2 k 2 -1 -1 k -1 b 2 k m z 1 2 4 b z 1 2 k m 4 b 2 k -1 m z 1 2 -4 4 2 m b z 1 2 b m -1 2 k z 1 2 -1 Binomial m k m 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 2 s -1 v g f z Binomial v s f 2 s -1 v -1 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 b b 2 2 k -1 m 2 4 f v -1 2 s -1 2 k -1 m 1 2 Erfi b 2 k -1 m 2 f v -1 2 s z 1 2 2 -1 f v -1 2 s 1 2 -1 -1 f v -1 2 s 3 2 -1 2 -1 m b z 1 2 2 k -1 m 2 g v -1 2 s f v -1 2 s z f v -1 2 s -1 -1 2 -1 b z 1 2 2 k -1 m -1 f v -1 2 s z f v -1 2 s -1 -1 1 f v -1 2 s 3 2 -1 -1 m b 2 g v -1 2 s -1 b 2 2 k -1 m 2 4 f v -1 2 s -1 2 k -1 m 1 2 Erfi b 2 k -1 m 2 f v -1 2 s z 1 2 2 f v -1 2 s 1 2 -1 -1 m -1 g v -1 2 s -1 b b 2 m -1 2 k 2 4 f v -1 2 s -1 m -1 2 k 1 2 Erfi b m -1 2 k 2 f v -1 2 s z 1 2 2 -1 f v -1 2 s 1 2 -1 -1 f v -1 2 s 3 2 -1 2 -1 m b z 1 2 m -1 2 k 2 g v -1 2 s f v -1 2 s z f v -1 2 s -1 -1 2 -1 b z 1 2 m -1 2 k -1 f v -1 2 s z f v -1 2 s -1 -1 1 f v -1 2 s 3 2 -1 -1 m b 2 g v -1 2 s -1 b 2 m -1 2 k 2 4 f v -1 2 s -1 m -1 2 k 1 2 Erfi b m -1 2 k 2 f v -1 2 s z 1 2 2 f v -1 2 s 1 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18