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 Cosh

 http://functions.wolfram.com/01.20.21.4217.01

 Input Form

 Integrate[Sinh[b Sqrt[z] + d z + e]^m Cosh[c Sqrt[z] + f z + g]^v, z] == (2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]))/I^m - 2^(-m - v) I^(1 - m) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/(I d (2 k - m))^(3/2)) (-1)^k Binomial[m, k] (b (2 k - m) Sqrt[2 Pi] (I Cosh[(b^2 (2 k - m))/(4 d) + e (-2 k + m) + (I m Pi)/2] FresnelC[(b (2 k - m) + 2 d (2 k - m) Sqrt[z])/ (Sqrt[I d (2 k - m)] Sqrt[2 Pi])] + FresnelS[(b (2 k - m) + 2 d (2 k - m) Sqrt[z])/(Sqrt[I d (2 k - m)] Sqrt[2 Pi])] Sinh[(b^2 (2 k - m))/(4 d) + e (-2 k + m) + (I m Pi)/2]) + 2 Sqrt[I d (2 k - m)] Sinh[e (-2 k + m) + (I m Pi)/2 - b (2 k - m) Sqrt[z] - d (2 k - m) z]), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-m - v) I^(1 - m) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(I f (-2 s + v))^(3/2)) Binomial[v, s] ((-c) Sqrt[2 Pi] (-2 s + v) (I Cosh[(c^2 (-2 s + v))/(4 f) - g (-2 s + v)] FresnelC[(c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/ (Sqrt[2 Pi] Sqrt[I f (-2 s + v)])] + FresnelS[(c (-2 s + v) + 2 f (-2 s + v) Sqrt[z])/ (Sqrt[2 Pi] Sqrt[I f (-2 s + v)])] Sinh[(c^2 (-2 s + v))/(4 f) - g (-2 s + v)]) + 2 Sqrt[I f (-2 s + v)] Sinh[g (-2 s + v) + c (-2 s + v) Sqrt[z] + f (-2 s + v) z]), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-m - v) I^(1 - m) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (((-Sqrt[2 Pi]) (2 b k - b m + 2 c s - c v) (I Cosh[2 e k - e m - (I m Pi)/2 + 2 g s - g v - (b (-2 k + m) + c (-2 s + v))^2/(8 d k - 4 d m + 8 f s - 4 f v)] FresnelC[(2 b k - b m + 2 c s - c v + 2 (2 d k - d m + 2 f s - f v) Sqrt[z])/(Sqrt[2 Pi] Sqrt[I (2 d k - d m + 2 f s - f v)])] - FresnelS[(2 b k - b m + 2 c s - c v + 2 (2 d k - d m + 2 f s - f v) Sqrt[z])/(Sqrt[2 Pi] Sqrt[I (2 d k - d m + 2 f s - f v)])] Sinh[2 e k - e m - (I m Pi)/2 + 2 g s - g v - (b (-2 k + m) + c (-2 s + v))^2/(8 d k - 4 d m + 8 f s - 4 f v)]) + 2 Sqrt[I (2 d k - d m + 2 f s - f v)] Sinh[2 e k - e m - (I m Pi)/2 + 2 g s - g v + (2 b k - b m + 2 c s - c v) Sqrt[z] + (2 d k - d m + 2 f s - f v) z])/ (I (2 d k - d m + 2 f s - f v))^(3/2) + ((-Sqrt[2 Pi]) (2 b k - b m - 2 c s + c v) (I Cosh[2 e k - e m - (I m Pi)/2 - 2 g s + g v - (2 b k - b m - 2 c s + c v)^2/(8 d k - 4 d m - 8 f s + 4 f v)] FresnelC[(2 b k - b m - 2 c s + c v + 2 (2 d k - d m - 2 f s + f v) Sqrt[z])/(Sqrt[2 Pi] Sqrt[I (2 d k - d m - 2 f s + f v)])] - FresnelS[(2 b k - b m - 2 c s + c v + 2 (2 d k - d m - 2 f s + f v) Sqrt[z])/(Sqrt[2 Pi] Sqrt[I (2 d k - d m - 2 f s + f v)])] Sinh[2 e k - e m - (I m Pi)/2 - 2 g s + g v - (2 b k - b m - 2 c s + c v)^2/(8 d k - 4 d m - 8 f s + 4 f v)]) + 2 Sqrt[I (2 d k - d m - 2 f s + f v)] Sinh[2 e k - e m - (I m Pi)/2 - 2 g s + g v + (2 b k - b m - 2 c s + c v) Sqrt[z] + (2 d k - d m - 2 f s + f v) z])/ (I (2 d k - d m - 2 f s + f v))^(3/2)), {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 ( z b + e + d z ) cosh v ( z c + g + f z ) z 2 - m - v - m 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 - 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]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 1 ( d ( 2 k - m ) ) 3 / 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]]]]] ( b ( 2 k - m ) 2 π ( cosh ( ( 2 k - m ) b 2 4 d + e ( m - 2 k ) + m π 2 ) C TagBox["C", FresnelC] ( b ( 2 k - m ) + 2 d z ( 2 k - m ) d ( 2 k - m ) 2 π ) + S TagBox["S", FresnelS] ( b ( 2 k - m ) + 2 d z ( 2 k - m ) d ( 2 k - m ) 2 π ) sinh ( ( 2 k - m ) b 2 4 d + e ( m - 2 k ) + m π 2 ) ) + 2 d ( 2 k - m ) sinh ( - d z ( 2 k - m ) - b z ( 2 k - m ) + e ( m - 2 k ) + m π 2 ) ) ) + 2 - m - v 1 - m ( 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 1 ( f ( v - 2 s ) ) 3 / 2 ( ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 f ( v - 2 s ) sinh ( g ( v - 2 s ) + f z ( v - 2 s ) + c z ( v - 2 s ) ) - c 2 π ( v - 2 s ) ( cosh ( c 2 ( v - 2 s ) 4 f - g ( v - 2 s ) ) C TagBox["C", FresnelC] ( c ( v - 2 s ) + 2 f z ( v - 2 s ) 2 π f ( v - 2 s ) ) + S TagBox["S", FresnelS] ( c ( v - 2 s ) + 2 f z ( v - 2 s ) 2 π f ( v - 2 s ) ) sinh ( c 2 ( v - 2 s ) 4 f - g ( v - 2 s ) ) ) ) ) + 2 - m - v 1 - 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]]]]] ( ( 2 ( 2 d k - d m - 2 f s + f v ) sinh ( 2 e k - e m - 2 g s + g v + ( 2 d k - d m - 2 f s + f v ) z + ( 2 b k - b m - 2 c s + c v ) z - m π 2 ) - 2 π ( 2 b k - b m - 2 c s + c v ) ( cosh ( - ( 2 b k - b m - 2 c s + c v ) 2 8 d k - 4 d m - 8 f s + 4 f v + 2 e k - e m - 2 g s + g v - m π 2 ) C TagBox["C", FresnelC] ( 2 b k - b m - 2 c s + c v + 2 ( 2 d k - d m - 2 f s + f v ) z 2 π ( 2 d k - d m - 2 f s + f v ) ) - S TagBox["S", FresnelS] ( 2 b k - b m - 2 c s + c v + 2 ( 2 d k - d m - 2 f s + f v ) z 2 π ( 2 d k - d m - 2 f s + f v ) ) sinh ( - ( 2 b k - b m - 2 c s + c v ) 2 8 d k - 4 d m - 8 f s + 4 f v + 2 e k - e m - 2 g s + g v - m π 2 ) ) ) / ( ( 2 d k - d m - 2 f s + f v ) ) 3 / 2 + ( 2 ( 2 d k - d m + 2 f s - f v ) sinh ( 2 e k - e m + 2 g s - g v + ( 2 d k - d m + 2 f s - f v ) z + ( 2 b k - b m + 2 c s - c v ) z - m π 2 ) - 2 π ( 2 b k - b m + 2 c s - c v ) ( cosh ( - ( b ( m - 2 k ) + c ( v - 2 s ) ) 2 8 d k - 4 d m + 8 f s - 4 f v + 2 e k - e m + 2 g s - g v - m π 2 ) C TagBox["C", FresnelC] ( 2 b k - b m + 2 c s - c v + 2 ( 2 d k - d m + 2 f s - f v ) z 2 π ( 2 d k - d m + 2 f s - f v ) ) - S TagBox["S", FresnelS] ( 2 b k - b m + 2 c s - c v + 2 ( 2 d k - d m + 2 f s - f v ) z 2 π ( 2 d k - d m + 2 f s - f v ) ) sinh ( - ( b ( m - 2 k ) + c ( v - 2 s ) ) 2 8 d k - 4 d m + 8 f s - 4 f v + 2 e k - e m + 2 g s - g v - m π 2 ) ) ) / ( ( 2 d k - d m + 2 f s - f v ) ) 3 / 2 ) /; m + v + Condition z z 1 2 b e d z m z 1 2 c g f z v 2 -1 m -1 v -1 m z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 -1 2 -1 m -1 v 1 -1 m Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 1 d 2 k -1 m 3 2 -1 -1 k Binomial m k b 2 k -1 m 2 1 2 2 k -1 m b 2 4 d -1 e m -1 2 k m 2 -1 FresnelC b 2 k -1 m 2 d z 1 2 2 k -1 m d 2 k -1 m 1 2 2 1 2 -1 FresnelS b 2 k -1 m 2 d z 1 2 2 k -1 m d 2 k -1 m 1 2 2 1 2 -1 2 k -1 m b 2 4 d -1 e m -1 2 k m 2 -1 2 d 2 k -1 m 1 2 -1 d z 2 k -1 m -1 b z 1 2 2 k -1 m e m -1 2 k m 2 -1 2 -1 m -1 v 1 -1 m Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 1 f v -1 2 s 3 2 -1 Binomial v s 2 f v -1 2 s 1 2 g v -1 2 s f z v -1 2 s c z 1 2 v -1 2 s -1 c 2 1 2 v -1 2 s c 2 v -1 2 s 4 f -1 -1 g v -1 2 s FresnelC c v -1 2 s 2 f z 1 2 v -1 2 s 2 1 2 f v -1 2 s 1 2 -1 FresnelS c v -1 2 s 2 f z 1 2 v -1 2 s 2 1 2 f v -1 2 s 1 2 -1 c 2 v -1 2 s 4 f -1 -1 g v -1 2 s 2 -1 m -1 v 1 -1 m k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s 2 2 d k -1 d m -1 2 f s f v 1 2 2 e k -1 e m -1 2 g s g v 2 d k -1 d m -1 2 f s f v z 2 b k -1 b m -1 2 c s c v z 1 2 -1 m 2 -1 -1 2 1 2 2 b k -1 b m -1 2 c s c v -1 2 b k -1 b m -1 2 c s c v 2 8 d k -1 4 d m -1 8 f s 4 f v -1 2 e k -1 e m -1 2 g s g v -1 m 2 -1 FresnelC 2 b k -1 b m -1 2 c s c v 2 2 d k -1 d m -1 2 f s f v z 1 2 2 1 2 2 d k -1 d m -1 2 f s f v 1 2 -1 -1 FresnelS 2 b k -1 b m -1 2 c s c v 2 2 d k -1 d m -1 2 f s f v z 1 2 2 1 2 2 d k -1 d m -1 2 f s f v 1 2 -1 -1 2 b k -1 b m -1 2 c s c v 2 8 d k -1 4 d m -1 8 f s 4 f v -1 2 e k -1 e m -1 2 g s g v -1 m 2 -1 2 d k -1 d m -1 2 f s f v 3 2 -1 2 2 d k -1 d m 2 f s -1 f v 1 2 2 e k -1 e m 2 g s -1 g v 2 d k -1 d m 2 f s -1 f v z 2 b k -1 b m 2 c s -1 c v z 1 2 -1 m 2 -1 -1 2 1 2 2 b k -1 b m 2 c s -1 c v -1 b m -1 2 k c v -1 2 s 2 8 d k -1 4 d m 8 f s -1 4 f v -1 2 e k -1 e m 2 g s -1 g v -1 m 2 -1 FresnelC 2 b k -1 b m 2 c s -1 c v 2 2 d k -1 d m 2 f s -1 f v z 1 2 2 1 2 2 d k -1 d m 2 f s -1 f v 1 2 -1 -1 FresnelS 2 b k -1 b m 2 c s -1 c v 2 2 d k -1 d m 2 f s -1 f v z 1 2 2 1 2 2 d k -1 d m 2 f s -1 f v 1 2 -1 -1 b m -1 2 k c v -1 2 s 2 8 d k -1 4 d m 8 f s -1 4 f v -1 2 e k -1 e m 2 g s -1 g v -1 m 2 -1 2 d k -1 d m 2 f s -1 f v 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