Solve $\int \frac{e^{x}(2-x^2)}{(1-x)\sqrt{1-x^2}}\mathrm dx$

Hint

Observe that the exponent of $1-x$ is $-\dfrac32$

So, let us find $$\dfrac{d\left(e^x\dfrac{(1+x)^n}{\sqrt{1-x}}\right)}{dx}$$

Compare with the given expression to find the value of $n$

Hint:

$$\dfrac{1+1-x^2}{(1-x)^{3/2}(1+x)^{1/2}}=\dfrac1{...}+f(x)$$

where $f(x)=\dfrac{\sqrt{1+x}}{\sqrt{1-x}},$

$f'(x)=?$

Recall $\dfrac{d(e^xf(x))}{dx}=?$

$x=\cos2t,dx=?$

$$-I=\int\dfrac{e^{\cos2t}(1+\sin^22t)}{\sin^2t}=e^{\cos2t}\csc^2t-\dfrac{d(e^{\cos2t})}{dt}(-\cot t)$$

$$=\dfrac{d(e^{\cos2t}(-\cot t))}{dt}$$