an integral problem $$
\int_{-\infty}^{\infty} [c_1 + c_2 (x-c_3)^2 + (x - c_4)]^{-c_5} \, dx
$$
with $c_1, c_2, c_3, c_4, c_5$ known real constant.
Can you help me to solve this integral?
 A: If we let 
$$
z = \frac{x+\frac{1-2c_2c_3}{2c_2}}{\sqrt{\frac{c_1+c_2c_3^2}{c_2}-\left(\frac{1-2c_2c_3}{2c_2}\right)^2}}
$$
(this was for brevity)
we then obtain
$$
\left[c_2\left(\frac{c_1+c_2c_3^2}{c_2}-\left(\frac{1-2c_2c_3}{2c_2}\right)^2\right)^{1-\frac{1}{2c_5}}\right]^{-c_5}\int_{-\infty}^{\infty}\left(z^2+1\right)^{-c_5}dz
$$
so now you have to integrate 
$$
\int_{-\infty}^{\infty}\left(z^2+1\right)^{-c_5}dz
$$
letting $z=\tan \theta$
we get
$$
\int_{-\infty}^{\infty}\left(z^2+1\right)^{-c_5}dz = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sec^{-2c_5}\theta \sec^2\theta \,d\theta
$$
if we have $c_5>1$
$$
\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sec^{-2c_5}\theta \sec^2\theta \,d\theta = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\cos^{2\left(c_5-1\right)}\theta \,d\theta
$$
or
if we have $c_5<1$
$$
\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sec^{-2c_5}\theta \sec^2\theta \,d\theta = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sec^{2\left(c_5-1\right)}\theta \,d\theta
$$
let me know if you can proceed..or if indeed I made a mistake!
