Calculate the given indefinite integral

Question

Calculate the following integral:

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

I believe I need to choose a good substitution, but the problem is that I haven't found it yet.

Thank you!

Answer 1

HINT:

Set $x=\tan y\implies dx=?$

$$\int \frac1{(1-x^2)\sqrt{1+x^2}} dx=\int\dfrac{\sec y\ dy}{1-\tan^2y}=\int\dfrac{\cos y\ dy}{1-2\sin^2y}$$

Set $\sin y=u$

Or directly, $u=\dfrac x{\sqrt{1+x^2}}$

Answer 2

HINT:

To evaluate integrals of the form $\displaystyle\int\frac{1}{(lx^2+m)\sqrt{ax^2+b}}\,dx,$ substitute $\sqrt{ax^2+b}=xz$ for another function $z$. This would lead to the primitive yielding an $\arctan$ function.