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TLDR: need help defining two functions involving integrals in the form of "Laplace Coefficents".

My intention is to define functions that involve Laplace Coefficents. In particular, if

$$b_{1/2}^j (x) = \frac{1}{\pi} \int\limits_0^{2\pi} \frac{\cos{(j\psi)}}{(1-2x \cos{\psi} + x^2)^{1/2}}d\psi$$

the task is to define operations with the terms $b_{1/2}^{q+1}(x)$ and $b_{1/2}^{q}(x)$. I've defined the needed as:

f1var[q_, \[Alpha]_] = (-(1/2))* (2*(q + 1)*(1/Pi)*Integrate[Cos[(q + 1)*\[Psi]]/(1 - 2*\[Alpha]*Cos[\[Psi]] + \[Alpha]^2)^(1/2), {\[Psi], 0, 2*Pi}] + \[Alpha]*D[(1/Pi)*Integrate[Cos[(q + 1)*\[Psi]]/(1 - 2*\[Alpha]*Cos[\[Psi]] + \[Alpha]^2)^(1/2), {\[Psi], 0, 2*Pi}], \[Alpha]])

or

$$\text{f1var}(\text{q$\_$},\alpha \_)=-\frac{1}{2} \left(\alpha \frac{\partial }{\partial \alpha }\frac{\int_0^{2 \pi } \frac{\cos ((q+1) \psi )}{\sqrt{\alpha ^2-2 \alpha \cos (\psi )+1}} \, d\psi }{\pi }+\frac{2 (q+1) \int_0^{2 \pi } \frac{\cos ((q+1) \psi )}{\sqrt{\alpha ^2-2 \alpha \cos (\psi )+1}} \, d\psi }{\pi }\right)$$ and

f2var[q_, \[Alpha]_] = (1/2)*((2*q + 1)*(1/Pi)*Integrate[Cos[q*\[Psi]]/(1 - 2*\[Alpha]*Cos[\[Psi]] + \[Alpha]^2)^(1/2), {\[Psi], 0, 2*Pi}] + \[Alpha]*D[(1/Pi)*Integrate[Cos[q*\[Psi]]/(1 - 2*\[Alpha]*Cos[\[Psi]] + \[Alpha]^2)^(1/2), {\[Psi], 0, 2*Pi}], \[Alpha]]) - 2*\[Alpha]

or

$$\text{f2var}(\text{q$\_$},\alpha \_)=\frac{1}{2} \left(\alpha \frac{\partial }{\partial \alpha }\frac{\int_0^{2 \pi } \frac{\cos (q \psi )}{\sqrt{\alpha ^2-2 \alpha \cos (\psi )+1}} \, d\psi }{\pi }+\frac{(2 q+1) \int_0^{2 \pi } \frac{\cos (q \psi )}{\sqrt{\alpha ^2-2 \alpha \cos (\psi )+1}} \, d\psi }{\pi }\right)-2 \alpha$$

The problem is that these functions don't stop running so I guess I'm doing something wrong. Question: why Mathematica is not defining the functions? Should I expand in series, for instance as this article suggest?

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