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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0054.01

 Input Form

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

 Standard Form

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 MathML Form

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