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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0068.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2001-10-29