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 Cosh

 http://functions.wolfram.com/01.20.21.4215.01

 Input Form

 Integrate[Sinh[b z^2 + d z + e]^m Cosh[c z^2 + 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^(1/2 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/Sqrt[I b (-2 k + m)]) (-1)^k Binomial[m, k] (I Cosh[(d^2 (-2 k + m))/(4 b) - e (-2 k + m) - (I m Pi)/2] FresnelC[(d (-2 k + m) + 2 b (-2 k + m) z)/(Sqrt[I b (-2 k + m)] Sqrt[2 Pi])] + FresnelS[(d (-2 k + m) + 2 b (-2 k + m) z)/ (Sqrt[I b (-2 k + m)] Sqrt[2 Pi])] Sinh[(d^2 (-2 k + m))/(4 b) - e (-2 k + m) - (I m Pi)/2]), {k, 0, Floor[(1/2) (-1 + m)]}])/I^m + (2^(1/2 - m - v) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/Sqrt[I c (-2 s + v)]) Binomial[v, s] (I Cosh[(f^2 (-2 s + v))/(4 c) - g (-2 s + v)] FresnelC[(f (-2 s + v) + 2 c (-2 s + v) z)/(Sqrt[2 Pi] Sqrt[I c (-2 s + v)])] + FresnelS[(f (-2 s + v) + 2 c (-2 s + v) z)/(Sqrt[2 Pi] Sqrt[I c (-2 s + v)])] Sinh[(f^2 (-2 s + v))/(4 c) - g (-2 s + v)]), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m + (2^(1/2 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((I Cosh[2 e k - e m - (I m Pi)/2 + 2 g s - g v - (d (-2 k + m) + f (-2 s + v))^2/(8 b k - 4 b m + 8 c s - 4 c v)] FresnelC[(d (2 k - m) + 2 b (2 k - m) z + (2 s - v) (f + 2 c z))/(Sqrt[2 Pi] Sqrt[I (2 b k - b m + 2 c s - c v)])] - FresnelS[(d (2 k - m) + 2 b (2 k - m) z + (2 s - v) (f + 2 c z))/(Sqrt[2 Pi] Sqrt[ I (2 b k - b m + 2 c s - c v)])] Sinh[2 e k - e m - (I m Pi)/ 2 + 2 g s - g v - (d (-2 k + m) + f (-2 s + v))^2/ (8 b k - 4 b m + 8 c s - 4 c v)])/Sqrt[2 I b k - I b m + 2 I c s - I c v] + (I Cosh[2 e k - e m - (I m Pi)/2 - 2 g s + g v - (2 d k - d m - 2 f s + f v)^2/(8 b k - 4 b m - 8 c s + 4 c v)] FresnelC[(2 d k - d m - 2 f s + f v + 2 (2 b k - b m - 2 c s + c v) z)/(Sqrt[2 Pi] Sqrt[I (2 b k - b m - 2 c s + c v)])] - FresnelS[(2 d k - d m - 2 f s + f v + 2 (2 b k - b m - 2 c s + c v) z)/(Sqrt[2 Pi] Sqrt[I (2 b k - b m - 2 c s + c v)])] Sinh[2 e k - e m - (I m Pi)/2 - 2 g s + g v - (2 d k - d m - 2 f s + f v)^2/(8 b k - 4 b m - 8 c s + 4 c v)])/ Sqrt[I (2 b k - b m - 2 c s + c v)]), {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

 sinh m ( b z 2 + d z + e ) cosh v ( c z 2 + f z + g ) 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 2 - m π ( 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 b ( m - 2 k ) ( ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( cosh ( ( m - 2 k ) d 2 4 b - e ( m - 2 k ) - m π 2 ) C TagBox["C", FresnelC] ( d ( m - 2 k ) + 2 b z ( m - 2 k ) b ( m - 2 k ) 2 π ) + S TagBox["S", FresnelS] ( d ( m - 2 k ) + 2 b z ( m - 2 k ) b ( m - 2 k ) 2 π ) sinh ( ( m - 2 k ) d 2 4 b - e ( m - 2 k ) - m π 2 ) ) ) + 2 - m - v + 1 2 - m π ( 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 c ( v - 2 s ) ( ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( cosh ( f 2 ( v - 2 s ) 4 c - g ( v - 2 s ) ) C TagBox["C", FresnelC] ( f ( v - 2 s ) + 2 c z ( v - 2 s ) 2 π c ( v - 2 s ) ) + S TagBox["S", FresnelS] ( f ( v - 2 s ) + 2 c z ( v - 2 s ) 2 π c ( v - 2 s ) ) sinh ( f 2 ( v - 2 s ) 4 c - g ( v - 2 s ) ) ) ) + 2 - m - v + 1 2 - m π 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]]]]] ( ( cosh ( - ( 2 d k - d m - 2 f s + f v ) 2 8 b k - 4 b m - 8 c s + 4 c v + 2 e k - e m - 2 g s + g v - m π 2 ) C TagBox["C", FresnelC] ( 2 d k - d m - 2 f s + f v + 2 ( 2 b k - b m - 2 c s + c v ) z 2 π ( 2 b k - b m - 2 c s + c v ) ) - S TagBox["S", FresnelS] ( 2 d k - d m - 2 f s + f v + 2 ( 2 b k - b m - 2 c s + c v ) z 2 π ( 2 b k - b m - 2 c s + c v ) ) sinh ( - ( 2 d k - d m - 2 f s + f v ) 2 8 b k - 4 b m - 8 c s + 4 c v + 2 e k - e m - 2 g s + g v - m π 2 ) ) / ( ( 2 b k - b m - 2 c s + c v ) ) + ( cosh ( - ( d ( m - 2 k ) + f ( v - 2 s ) ) 2 8 b k - 4 b m + 8 c s - 4 c v + 2 e k - e m + 2 g s - g v - m π 2 ) C TagBox["C", FresnelC] ( d ( 2 k - m ) + 2 b z ( 2 k - m ) + ( 2 s - v ) ( f + 2 c z ) 2 π ( 2 b k - b m + 2 c s - c v ) ) - S TagBox["S", FresnelS] ( d ( 2 k - m ) + 2 b z ( 2 k - m ) + ( 2 s - v ) ( f + 2 c z ) 2 π ( 2 b k - b m + 2 c s - c v ) ) sinh ( - ( d ( m - 2 k ) + f ( v - 2 s ) ) 2 8 b k - 4 b m + 8 c s - 4 c v + 2 e k - e m + 2 g s - g v - m π 2 ) ) / ( 2 b k - b m + 2 c s - c v ) ) /; m + v + Condition z b z 2 d z e m c z 2 f z g 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 2 -1 m -1 v 1 2 -1 m 1 2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 1 b m -1 2 k 1 2 -1 -1 k Binomial m k m -1 2 k d 2 4 b -1 -1 e m -1 2 k -1 m 2 -1 FresnelC d m -1 2 k 2 b z m -1 2 k b m -1 2 k 1 2 2 1 2 -1 FresnelS d m -1 2 k 2 b z m -1 2 k b m -1 2 k 1 2 2 1 2 -1 m -1 2 k d 2 4 b -1 -1 e m -1 2 k -1 m 2 -1 2 -1 m -1 v 1 2 -1 m 1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 1 c v -1 2 s 1 2 -1 Binomial v s f 2 v -1 2 s 4 c -1 -1 g v -1 2 s FresnelC f v -1 2 s 2 c z v -1 2 s 2 1 2 c v -1 2 s 1 2 -1 FresnelS f v -1 2 s 2 c z v -1 2 s 2 1 2 c v -1 2 s 1 2 -1 f 2 v -1 2 s 4 c -1 -1 g v -1 2 s 2 -1 m -1 v 1 2 -1 m 1 2 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 2 d k -1 d m -1 2 f s f v 2 8 b k -1 4 b m -1 8 c s 4 c v -1 2 e k -1 e m -1 2 g s g v -1 m 2 -1 FresnelC 2 d k -1 d m -1 2 f s f v 2 2 b k -1 b m -1 2 c s c v z 2 1 2 2 b k -1 b m -1 2 c s c v 1 2 -1 -1 FresnelS 2 d k -1 d m -1 2 f s f v 2 2 b k -1 b m -1 2 c s c v z 2 1 2 2 b k -1 b m -1 2 c s c v 1 2 -1 -1 2 d k -1 d m -1 2 f s f v 2 8 b k -1 4 b m -1 8 c s 4 c v -1 2 e k -1 e m -1 2 g s g v -1 m 2 -1 2 b k -1 b m -1 2 c s c v 1 2 -1 -1 d m -1 2 k f v -1 2 s 2 8 b k -1 4 b m 8 c s -1 4 c v -1 2 e k -1 e m 2 g s -1 g v -1 m 2 -1 FresnelC d 2 k -1 m 2 b z 2 k -1 m 2 s -1 v f 2 c z 2 1 2 2 b k -1 b m 2 c s -1 c v 1 2 -1 -1 FresnelS d 2 k -1 m 2 b z 2 k -1 m 2 s -1 v f 2 c z 2 1 2 2 b k -1 b m 2 c s -1 c v 1 2 -1 -1 d m -1 2 k f v -1 2 s 2 8 b k -1 4 b m 8 c s -1 4 c v -1 2 e k -1 e m 2 g s -1 g v -1 m 2 -1 2 b k -1 b m 2 c s -1 c v 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