Hyperbolic sine: Integration (formula 01.19.21.2226)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving functions of the direct function and exponential function

Involving products of powers of two direct functions and exponential function

Involving product of power of the direct function, the direct function and rational functions of exp

Involving sinh(e z)sinh^v(c z)(a+b e^{d z})^-n

http://functions.wolfram.com/01.19.21.2226.01

Input Form

Integrate[(Sinh[e z] Sinh[c z]^v)/(a + b E^(d z))^n, z] == (1/e) (((I/2)^(1 + v) Binomial[v, v/2] ((-E^((I Pi)/2 - e z)) Hypergeometric2F1[-(e/d), n, (d - e)/d, -((b E^(d z))/a)] + E^(-((I Pi)/2) + e z) Hypergeometric2F1[e/d, n, (d + e)/d, -((b E^(d z))/a)]) (1 - Mod[v, 2]))/a^n) + ((I/2)^(1 + v) Sum[(-1)^s Binomial[v, s] ((1/(-e - 2 c s + c v)) (Cos[(1/2) Pi (1 - v)] ((-E^((e + 2 c s - c v) z)) Hypergeometric2F1[(e + 2 c s - c v)/d, n, (d + e + 2 c s - c v)/d, -((b E^(d z))/a)] + E^((-e - 2 c s + c v) z) Hypergeometric2F1[(-e - 2 c s + c v)/d, n, (d - e - 2 c s + c v)/d, -((b E^(d z))/a)])) + (1/(-e + 2 c s - c v)) (Cos[(1/2) Pi (1 + v)] (E^((-e + 2 c s - c v) z) Hypergeometric2F1[(-e + 2 c s - c v)/d, n, (d - e + 2 c s - c v)/d, -((b E^(d z))/a)] - E^((e - 2 c s + c v) z) Hypergeometric2F1[(e - 2 c s + c v)/d, n, (d + e - 2 c s + c v)/d, -((b E^(d z))/a)])) + (1/(-e - 2 c s + c v)) (I (E^((e + 2 c s - c v) z) Hypergeometric2F1[ (e + 2 c s - c v)/d, n, (d + e + 2 c s - c v)/d, -((b E^(d z))/a)] + E^((-e - 2 c s + c v) z) Hypergeometric2F1[ (-e - 2 c s + c v)/d, n, (d - e - 2 c s + c v)/d, -((b E^(d z))/a)]) Sin[(1/2) Pi (1 - v)]) + (1/(-e + 2 c s - c v)) (I (E^((-e + 2 c s - c v) z) Hypergeometric2F1[(-e + 2 c s - c v)/d, n, (d - e + 2 c s - c v)/ d, -((b E^(d z))/a)] + E^((e - 2 c s + c v) z) Hypergeometric2F1[(e - 2 c s + c v)/d, n, (d + e - 2 c s + c v)/ d, -((b E^(d z))/a)]) Sin[(1/2) Pi (1 + v)])), {s, 0, Floor[(1/2) (-1 + v)]}])/a^n /; Element[n, Integers] && n > 0 && Element[v, Integers] && v > 0

Standard Form

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MathML Form

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</mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List["-", "e"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], "d"], Hypergeometric2F1], ",", TagBox["n", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["d", "-", "e", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", 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Rule Form

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

2002-12-18