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 Cosh

 http://functions.wolfram.com/01.20.21.1233.01

 Input Form

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

 Standard Form

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 MathML Form

 b z 2 + d z + e sin m ( a z 2 + p z + q ) cosh ( c z 2 + f z + g ) z 2 - m - 2 π ( 1 ( b - c ) ( b + c ) ( ( 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]]]]] ( b - c ( b + c ) - ( d - f ) 2 4 ( b - c ) + e - g erfi ( d - f + 2 b z - 2 c z 2 b - c ) + ( b - c ) b + c - ( d + f ) 2 4 ( b + c ) + e + g erfi ( d + f + 2 ( b + c ) z 2 b + c ) ) ( 1 - m mod 2 \$CellContext`m 2 ) ) + 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]]]]] ( ( - ( d + f + ( 2 k - m ) p ) 2 4 ( b + c + a ( 2 k - m ) ) + e + g + ( 2 k - m ) q + m π 2 b + c + a ( 2 k - m ) ( b - c - 2 a k + a m ) erfi ( d + f + ( 2 k - m ) p + 2 ( b + c + a ( 2 k - m ) ) z 2 b + c + a ( 2 k - m ) ) + ( - d + f + ( m - 2 k ) p ) 2 4 ( b - c - 2 a k + a m ) + e - g - ( 2 k - m ) q - m π 2 ( b + c + a ( 2 k - m ) ) b - c - 2 a k + a m erfi ( d - f - 2 k p + m p + 2 b z - 2 c z - 4 a k z + 2 a m z 2 b - c - 2 a k + a m ) ) / ( ( b + c + a ( 2 k - m ) ) ( b - c - 2 a k + a m ) ) + ( - ( d + f + ( m - 2 k ) p ) 2 4 ( b + c + a ( m - 2 k ) ) + e + g - ( 2 k - m ) q - m π 2 b + c + a ( m - 2 k ) ( b - c - a ( m - 2 k ) ) erfi ( d + f + ( m - 2 k ) p + 2 ( b + c + a ( m - 2 k ) ) z 2 b + c + a ( m - 2 k ) ) + - ( - d + f + ( m - 2 k ) p ) 2 4 ( b - c - a ( m - 2 k ) ) + e - g + ( 2 k - m ) q + m π 2 ( b + c + a ( m - 2 k ) ) b - c - a ( m - 2 k ) erfi ( d - f - ( m - 2 k ) p + 2 b z - 2 ( c + a ( m - 2 k ) ) z 2 b - c - a ( m - 2 k ) ) ) / ( ( b - c - a ( m - 2 k ) ) ( b + c + a ( m - 2 k ) ) ) ) ) /; m + Condition z b z 2 d z e a z 2 p z q m c z 2 f z g 2 -1 m -2 1 2 1 b -1 c b c -1 Binomial m m 2 -1 b -1 c 1 2 b c -1 d -1 f 2 4 b -1 c -1 e -1 g Erfi d -1 f 2 b z -1 2 c z 2 b -1 c 1 2 -1 b -1 c b c 1 2 -1 d f 2 4 b c -1 e g Erfi d f 2 b c z 2 b c 1 2 -1 1 -1 \$CellContext`m 2 k 0 m -1 2 -1 -1 k Binomial m k -1 d f 2 k -1 m p 2 4 b c a 2 k -1 m -1 e g 2 k -1 m q m 2 -1 b c a 2 k -1 m 1 2 b -1 c -1 2 a k a m Erfi d f 2 k -1 m p 2 b c a 2 k -1 m z 2 b c a 2 k -1 m 1 2 -1 -1 d f m -1 2 k p 2 4 b -1 c -1 2 a k a m -1 e -1 g -1 2 k -1 m q -1 m 2 -1 b c a 2 k -1 m b -1 c -1 2 a k a m 1 2 Erfi d -1 f -1 2 k p m p 2 b z -1 2 c z -1 4 a k z 2 a m z 2 b -1 c -1 2 a k a m 1 2 -1 b c a 2 k -1 m b -1 c -1 2 a k a m -1 -1 d f m -1 2 k p 2 4 b c a m -1 2 k -1 e g -1 2 k -1 m q -1 m 2 -1 b c a m -1 2 k 1 2 b -1 c -1 a m -1 2 k Erfi d f m -1 2 k p 2 b c a m -1 2 k z 2 b c a m -1 2 k 1 2 -1 -1 -1 d f m -1 2 k p 2 4 b -1 c -1 a m -1 2 k -1 e -1 g 2 k -1 m q m 2 -1 b c a m -1 2 k b -1 c -1 a m -1 2 k 1 2 Erfi d -1 f -1 m -1 2 k p 2 b z -1 2 c a m -1 2 k z 2 b -1 c -1 a m -1 2 k 1 2 -1 b -1 c -1 a m -1 2 k b c a m -1 2 k -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18