Evaluating $\int \frac{1}{(1-x^2)^{3/2}} dx$

Question

I am struggling to evaluate the following integral:
$$\int \frac{1}{(1-x^2)^{3/2}} dx$$
I tried a lot to factorize the expression but I didn't reach the solution. Please someone help me.

Answer 1

Hint:

Set $x=\sin(t)$, then everything will turn out very well.

This often helps when you have some expresion like $1-x^2$ in your integral.

Answer 2

$$\int \frac{dx}{\left(1-x^2\right)^{3/2}}=[x=\sin t]=\int\frac{\cos t dt}{\cos^3 t}=\int\frac{dt}{\cos^2 t}=\tan t$$