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http://functions.wolfram.com/01.19.21.2159.01
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Integrate[E^(d z + e) Sinh[c Sqrt[z] + f z + g]^v, z] ==
((I/2)^v E^(e + d z) Binomial[v, v/2] (1 - Mod[v, 2]))/d +
Sum[(-1)^k Binomial[v, k]
(((-1)^v E^(e + 2 g k - g v + (2 c k - c v) Sqrt[z] +
(d + 2 f k - f v) z))/(d + 2 f k - f v) +
E^(e - 2 g k + g v + (-2 c k + c v) Sqrt[z] + (d - 2 f k + f v) z)/
(d - 2 f k + f v) - ((-1)^v E^(e + 2 g k - g v - (2 c k - c v)^2/
(4 (d + 2 f k - f v))) Sqrt[Pi] (2 c k - c v)
Erfi[(c (2 k - v) + 2 (d + 2 f k - f v) Sqrt[z])/
(2 Sqrt[d + 2 f k - f v])])/(2 (d + 2 f k - f v)^(3/2)) -
(c E^(e - 2 g k + g v - (c^2 (-2 k + v)^2)/(4 (d + f (-2 k + v))))
Sqrt[Pi] (-2 k + v) Erfi[(c (-2 k + v) + 2 (d - 2 f k + f v)
Sqrt[z])/(2 Sqrt[d + f (-2 k + v)])])/
(2 (d - 2 f k + f v)^(3/2))), {k, 0, Floor[(1/2) (-1 + v)]}]/2^v /;
Element[v, Integers] && v > 0
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mi> v </mi> </msup> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mi> g </mi> <mo> + </mo> <mrow> <mi> f </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> v </mi> </mtd> </mtr> <mtr> <mtd> <mfrac> <mi> v </mi> <mn> 2 </mn> </mfrac> </mtd> </mtr> </mtable> 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