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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0026.01

 Input Form

 Integrate[z^n Cos[b z] SinhIntegral[a z], z] == (1/4) (I b)^(-1 - n) n! (ExpIntegralEi[(a - I b) z] - ExpIntegralEi[(-(a + I b)) z] - ExpIntegralEi[(a + I b) z]/(-1)^n + (-1)^n ExpIntegralEi[(-a) z + I b z] + (1/Gamma[2 + n]) (2 (1 + n) ((-1)^n Gamma[1 + n, (-I) b z] - Gamma[1 + n, I b z]) SinhIntegral[a z]) + (-1)^n E^((a + I b) z) Sum[(((I b)/(a + I b))^m Sum[((-a - I b)^k z^k)/k!, {k, 0, -1 + m}])/m, {m, 1, n}] - (-1)^n E^((-a) z + I b z) Sum[(((I b)/(I b - a))^m Sum[((a - I b)^k z^k)/k!, {k, 0, -1 + m}])/m, {m, 1, n}] - E^((a - I b) z) Sum[(((I b)/(I b - a))^m Sum[((-a + I b)^k z^k)/k!, {k, 0, -1 + m}])/m, {m, 1, n}] + Sum[(((I b)/(I b + a))^m Sum[((a + I b)^k z^k)/k!, {k, 0, -1 + m}])/m, {m, 1, n}]/E^((a + I b) z)) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n cos ( b z ) Shi ( a z ) z 1 4 ( b ) - n - 1 n ! ( - Ei ( - ( a + b ) z ) - ( - 1 ) - n Ei ( ( a + b ) z ) + Ei ( ( a - b ) z ) + ( - 1 ) n Ei ( b z - a z ) + 1 Γ ( n + 2 ) ( 2 ( n + 1 ) ( ( - 1 ) n Γ ( n + 1 , - b z ) - Γ ( n + 1 , b z ) ) Shi ( a z ) ) - ( a - b ) z m = 1 n 1 m ( b b - a ) m k = 0 m - 1 ( b - a ) k z k k ! + - ( a + b ) z m = 1 n 1 m ( b a + b ) m k = 0 m - 1 ( a + b ) k z k k ! + ( - 1 ) n ( a + b ) z m = 1 n 1 m ( b a + b ) m k = 0 m - 1 ( - a - b ) k z k k ! - ( - 1 ) n b z - a z m = 1 n 1 m ( b b - a ) m k = 0 m - 1 ( a - b ) k z k k ! ) /; n Condition z z n b z SinhIntegral a z 1 4 b -1 n -1 n -1 ExpIntegralEi -1 a b z -1 -1 -1 n ExpIntegralEi a b z ExpIntegralEi a -1 b z -1 n ExpIntegralEi b z -1 a z 1 Gamma n 2 -1 2 n 1 -1 n Gamma n 1 -1 b z -1 Gamma n 1 b z SinhIntegral a z -1 a -1 b z m 1 n 1 m -1 b b -1 a -1 m k 0 m -1 b -1 a k z k k -1 -1 a b z m 1 n 1 m -1 b a b -1 m k 0 m -1 a b k z k k -1 -1 n a b z m 1 n 1 m -1 b a b -1 m k 0 m -1 -1 a -1 b k z k k -1 -1 -1 n b z -1 a z m 1 n 1 m -1 b b -1 a -1 m k 0 m -1 a -1 b k z k k -1 n [/itex]

 Rule Form

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

 2001-10-29