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I am searching the indefinite integral of this function: $\dfrac{\exp(x)}{(1+x)^{5/3}}$. Thank you alot.

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closed as off-topic by GNUSupporter 8964民主女神 地下教會, Lord Shark the Unknown, José Carlos Santos, Vinyl_cape_jawa, Song Feb 28 at 15:22

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    $\begingroup$ First thing I would do is make a change if variable from $x+1=u$ $\endgroup$ – Chinny84 Jul 25 '14 at 16:05
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    $\begingroup$ It cannot be expressed in terms of elementary functions. See exponential integral and/or incomplete $\Gamma$ function for more details. $\endgroup$ – Lucian Jul 25 '14 at 18:15
  • $\begingroup$ I have tried integration by parts, twice after that I find the integral of exp(x^3) dx. any ideas? $\endgroup$ – Ahmed Abrous Jul 25 '14 at 18:22
  • $\begingroup$ You are looking for the holy grail since, as Lucian just said, functions like $e^{x^n}$ (with $n\geq 2$) do not have an "elementary" antiderivative. $\endgroup$ – Jack D'Aurizio Jul 26 '14 at 2:13
  • $\begingroup$ Are you looking for a primitive or for the value of a specific integral of this function? $\endgroup$ – Did Jul 31 '14 at 20:52
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This integral is not elementary:
First rewrite your integral: $$\int{\frac{e^x}{(x+1)^{5/3}}}{dx} = e^{-1}\int{\frac{e^{x+1}}{(x+1)^{5/3}}}{dx}$$ Make a substitution $u = x+1$ and $du = dx$ $$e^{-1}\int{\frac{e^{x+1}}{(x+1)^{5/3}}}{dx} = e^{-1}\int{\frac{e^{u}}{u^{5/3}}}{du}$$ Make another substitution: $s = u^{1/3}$ and $du = 3s^2ds$ $$e^{-1}\int{\frac{e^{u}}{u^{5/3}}}{du} = 3e^{-1}\int{\frac{e^{s^3}}{s^3}}{ds}$$ The last integral is not elementary (which can be proven by the Risch Algorithm). Thus you can conclude that your initial integral is not an elementary function.

However, your integral has a closed form in terms of special functions (using Mathematica): $$\int_{-\infty}^{\infty}{s(\omega)e^{\tau*I*\omega}}{d\omega} = \frac{\left(\frac{2}{3}\right)^{2/3} \pi \text{c1} |r|^{2/3} \left(-i \text{sgn}(r) \left(3 \left(\sqrt[3]{-\text{c2}}-\sqrt[3]{\text{c2}}\right) \cos \left(\frac{2 |r|}{3 \text{c2}}\right)-\sqrt{3} \left(\sqrt[3]{-\text{c2}}+\sqrt[3]{\text{c2}}\right) \sin \left(\frac{2 |r|}{3 \text{c2}}\right)\right)+3 \left(\sqrt[3]{\text{c2}}-\sqrt[3]{-\text{c2}}\right) \sin \left(\frac{2 |r|}{3 \text{c2}}\right)-\sqrt{3} \left(\sqrt[3]{-\text{c2}}+\sqrt[3]{\text{c2}}\right) \cos \left(\frac{2 |r|}{3 \text{c2}}\right)\right)}{3 \text{c2}^2 \Gamma \left(\frac{2}{3}\right)}$$

with the restrictions: $r \in \mathbb{R} \land c2 \in \mathbb{C}\backslash\mathbb{R} \lor c2 = 0$

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  • $\begingroup$ Can I find the definite value of this integral numerically? $\endgroup$ – Ahmed Abrous Jul 26 '14 at 18:07
  • $\begingroup$ Of course. But you need to specify the constants first. What do you mean with $-\infty < \tau <\infty$? Is this a double integral? $\endgroup$ – Andrei Kh Aug 5 '14 at 20:34

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