How might you sort these integrals into different types?

\int \frac{1}{1 + x^{2}} dx \int \frac{1}{1 - x^{2}} dx

\int \frac{1}{(1 + x)^{2}} dx \int \frac{1}{(1 - x)^{2}} dx

\int \frac{1}{1 + x} dx \int \frac{1}{1 - x} dx

\int \frac{1}{\sqrt{1 + x}} dx \int \frac{1}{\sqrt{1 - x}} dx

\int \sqrt{1 + x^{2}} dx \int \sqrt{1 - x^{2}} dx

\int \sqrt{1 + x} dx \int \sqrt{1 - x} dx

How might you arrange the integrals if in each case we make the alteration

\int f (x) dx \to \int f (\sin (x)) dx