Hyperbolic sine integral: Integration (formula 06.39.21.0065)

SinhIntegral

$Mathematica Notation$

Gamma, Beta, Erf

SinhIntegral[z]

Integration

Indefinite integration

Involving direct function and Gamma-, Beta-, Erf-type functions

Involving exponential integral-type functions and a power function

Involving Ei and power

http://functions.wolfram.com/06.39.21.0065.01

Input Form

Integrate[z ExpIntegralEi[b z] SinhIntegral[a z], z] == (1/(4 a^2 b^2)) (a b E^((-a + b) z) + a b E^((a + b) z) + (a^2 + b^2) ExpIntegralEi[(-a + b) z] - a^2 ExpIntegralEi[(a + b) z] - b^2 ExpIntegralEi[(a + b) z] + b^2 ExpIntegralEi[b z] Gamma[2, (-a) z] - b^2 ExpIntegralEi[b z] Gamma[2, a z] + 2 a^2 (b^2 z^2 ExpIntegralEi[b z] + Gamma[2, (-b) z]) SinhIntegral[a z])

Standard Form

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MathML Form

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</mo> <mi> b </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Shi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - 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Rule Form

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

2001-10-29