To solve the integral

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

we can proceed by using trigonometric substitution.

Step 1: Use the substitution

Let $x = \sin \theta$. Then, $dx = \cos \theta \, d\theta$, and the integral becomes:

$\int \frac{1}{(1+\sin^2 \theta)\sqrt{1-\sin^2 \theta}} \cos \theta \, d\theta.$

Step 2: Simplify the expression

Since $\sqrt{1 - \sin^2 \theta} = \cos \theta$, the integral simplifies to:

$\int \frac{\cos \theta}{(1+\sin^2 \theta)\cos \theta} \, d\theta.$

This reduces further to:

$\int \frac{1}{1 + \sin^2 \theta} \, d\theta.$

Step 3: Use the Weierstrass substitution or identity

We can use the identity $\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}$ to rewrite the integrand:

$\int \frac{1}{1 + \frac{1 - \cos(2\theta)}{2}} \, d\theta = \int \frac{1}{\frac{3 + \cos(2\theta)}{2}} \, d\theta.$

This simplifies to:

$\int \frac{2}{3 + \cos(2\theta)} \, d\theta.$

Step 4: Solve the integral

The integral $\int \frac{1}{a + b\cos(2\theta)} \, d\theta$ is a standard form. Using the formula:

[ \int \frac{d