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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0038.01

 Input Form

 Integrate[z^n Cosh[b z] SinhIntegral[a z], z] == ((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 cosh ( b z ) Shi ( a z ) z 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 Shi ( a z ) - b 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 b z SinhIntegral a z 1 4 n b -1 n -1 -1 ExpIntegralEi -1 a -1 b z 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 -1 2 SinhIntegral a z -1 b 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 -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 -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