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 Cosh

 http://functions.wolfram.com/01.20.21.0847.01

 Input Form

 Integrate[Cos[b Sqrt[z] + d z + e]^m Cosh[c Sqrt[z] + f z + g], z] == ((-2^(-2 - m)) Binomial[m, m/2] (-((4 E^g Sinh[g + c Sqrt[z] + f z])/f) + (c E^(c^2/(4 f)) Sqrt[Pi] Erfi[(c + 2 f Sqrt[z])/(2 Sqrt[-f])])/ (-f)^(3/2) + (c E^(-(c^2/(4 f)) + 2 g) Sqrt[Pi] Erfi[(c + 2 f Sqrt[z])/(2 Sqrt[f])])/f^(3/2)) (1 - Mod[m, 2]))/E^g + 2^(-2 - m) Sum[Binomial[m, k] (E^(-g - I e (2 k - m)) ((2 E^(2 (g + I e (2 k - m)) + (c + I b (2 k - m)) Sqrt[z] + (f + I d (2 k - m)) z))/(f + I d (2 k - m)) + (2 E^((-c - I b (2 k - m)) Sqrt[z] + (-f + I d (-2 k + m)) z))/ (-f + I d (-2 k + m)) - (E^(-((c + I b (2 k - m))^2/(4 (f + I d (2 k - m)))) + 2 (g + I e (2 k - m))) (c + I b (2 k - m)) Sqrt[Pi] Erfi[(c + I b (2 k - m) + 2 (f + I d (2 k - m)) Sqrt[z])/ (2 Sqrt[f + I d (2 k - m)])])/(f + I d (2 k - m))^(3/2) - ((c + I b (2 k - m)) Sqrt[Pi] Erfi[(c + I b (2 k - m) + 2 (f + I d (2 k - m)) Sqrt[z])/(2 Sqrt[-f + I d (-2 k + m)])])/ E^((-c - I b (2 k - m))^2/(4 (-f + I d (-2 k + m))))/ (-f + I d (-2 k + m))^(3/2)) + E^(-g - I e (-2 k + m)) ((2 E^((-c - I b (-2 k + m)) Sqrt[z] + (-f + I d (2 k - m)) z))/ (-f + I d (2 k - m)) + (2 E^(2 (g + I e (-2 k + m)) + (c + I b (-2 k + m)) Sqrt[z] + (f + I d (-2 k + m)) z))/ (f + I d (-2 k + m)) - ((c + I b (-2 k + m)) Sqrt[Pi] Erfi[(c + I b (-2 k + m) + 2 (f + I d (-2 k + m)) Sqrt[z])/ (2 Sqrt[-f + I d (2 k - m)])])/E^((-c - I b (-2 k + m))^2/ (4 (-f + I d (2 k - m))))/(-f + I d (2 k - m))^(3/2) - (E^(-((c + I b (-2 k + m))^2/(4 (f + I d (-2 k + m)))) + 2 (g + I e (-2 k + m))) (c + I b (-2 k + m)) Sqrt[Pi] Erfi[(c + I b (-2 k + m) + 2 (f + I d (-2 k + m)) Sqrt[z])/ (2 Sqrt[f + I d (-2 k + m)])])/(f + I d (-2 k + m))^(3/2))), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18