I'm looking for a solution to the following integral. However, it seems it doesn't have a solution.


$$\int\limits\_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{\displaystyle1}{\displaystyle \epsilon+3}} y^{-\frac{\displaystyle\epsilon+4}{\displaystyle\epsilon+3}}dy,$$


where $\theta \ge 1$, $\epsilon \ge 0$, $x > \theta -1$ and it depends on a number of other parameters.


This equation appears in the context of Physical Layer Security, which is an area of study in digital communications (telecom).



Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?
With some effort (the lower integration limit requires care) I found this answer for the definite integral:


$$I(x)=\int\limits\_{\delta}^{x} \left(y-\delta\right)^{-\frac{1}{ \epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy=\frac{ (-2/\delta)^{\frac{2}{{\varepsilon}+3}} \pi ^{3/2} }{\sin \left(\pi\frac{{\varepsilon}+2}{{\varepsilon}+3}\right)\Gamma \left(\frac{{\varepsilon}+1}{2 {\varepsilon}+6}\right) \Gamma \left(\frac{\varepsilon+4}{{\varepsilon}+3}\right)}$$
$$\qquad\qquad+\;\delta^{-1}({\varepsilon}+3) x^{-\frac{1}{{\varepsilon}+3}} (x-{\delta})^{\frac{{\varepsilon}+2}{{\varepsilon}+3}} \, \_2F\_1\left(1,\frac{{\varepsilon}+1}{{\varepsilon}+3};\frac{{\varepsilon}+2}{{\varepsilon}+3};\frac{x}{{\delta}}\right),$$
for $\delta\equiv \theta-1>0$ and $x>\delta$, $\varepsilon>0$. Each of the two terms on the right-hand-side is complex, but the imaginary parts cancel.




---


Here is a Mathematica command to test this result against a numerical evaluation of the integral:



> 
> Plot[{((-4)^(1/(3 + eps)) delta^(-(2/(3 + eps))) [Pi]^(3/2)
> Csc[((2 + eps) [Pi])/(3 + eps)])/( Gamma[(1 + eps)/(6 + 2 eps)] Gamma[
> 1 + 1/(3 + eps)]) + ((3 + eps) x^(-(1/(3 + eps))) (-delta + x)^((
> 2 + eps)/(3 + eps))
> Hypergeometric2F1[1, (1 + eps)/(3 + eps), (2 + eps)/(3 + eps), x/
> delta])/delta,  
> 
> NIntegrate[ y^(-(eps + 4)/(eps + 3))\*(y - delta)^(-1/(eps + 3)), {y, delta,
> x}]}, {x, delta, 2}]
> 
> 
>
with this output for $\varepsilon=0.2,\delta=0.3$ (two indistinguishable curves)


![](https://i.stack.imgur.com/7LYTD.png)
