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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0063.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2001-10-29