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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0072.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2001-10-29