Pseudo-elliptic integrals

Integrals $\int f(x,y)\, dx$, where $f(x,y)$ is a rational function and $y=\sqrt{Ax^{4}+Bx^{3}+Cx^{2}+Dx+E}$, called elliptic integrals, generally cannot be integrated in closed form unless there are special relations among the coefficients of the function $f(x,y)$ or of the polynomial $Ax^{4}+Bx^{3}+Cx^{2}+Dx+E$.

In the following examples with palindromic polynomials, one of the substitutions $x+\frac{1}{x}=u$ or $x-\frac{1}{x}=v$ makes it comparatively easy to obtain the integral. Find the integral $\int \frac{x^{2}-1}{x^{2}+1}\frac{dx}{\sqrt{x^{4}+1}}$.

Find the integral $\int \frac{x^{2}+1}{x^{2}-1}\frac{dx}{\sqrt{x^{4}+1}}$.

Find the integral $\int \frac{x^{2}-1}{x\sqrt{1+3x^{2}+x^{4}}}\, dx$.

Find the integral $\int \frac{(x^{2}+1)\, dx}{x\sqrt{x^{4}+3x^{3}-2x^{2}-3x+1}}$.

Find the integral $\int \frac{\sqrt{1+x^{4}}}{1-x^{4}}\, dx$.

Find the integral $\int \frac{x+1}{x-1}\frac{dx}{\sqrt{x(x^{2}+x+1)}}$.

Find the integral $\int \frac{x+1}{x-2}\frac{dx}{\sqrt{1+x^{3}}}$.

Find the integral $\int \frac{(x^{2}-a^{2})\, dx}{x\sqrt{a^{4}+x^{4}}}$.

Find the integral $\int \frac{dx}{(1+x^{4})\sqrt{\sqrt{1+x^{4}}-x^{2}}}$.

Find the integral $\int \frac{dx}{\sqrt{(x-2)(2-3x+x^{3})}}$.

Find the pseudo-elliptic integral: