<s>[INST] Indefinite integration : $\int \frac{1+x-x^2}{\sqrt{(1-x^2)^3}}$ Problem : 
Solve : $\int \frac{1+x-x^2}{\sqrt{(1-x^2)^3}}$
I tried : 
$\frac{1-x^2}{\sqrt{(1-x^2)^3}} + \frac{x}{\sqrt{(1-x^2)^3}}$
But it's not working....Please guide how to proceed 
 [/INST] What you did is a very good first step. 
For the second integral, make the substitution $u=1-x^2$.  We end up with the easy $\int -\frac{1}{2}u^{-3/2}\,du $. 
For the first integral, note that the bottom simplifies to $(1-x^2)\sqrt{1-x^2}$, since the square root is only defined when $|x|\le 1$. There is nice cancellation, and we are integrating $\frac{1}{\sqrt{1-x^2}}$, easy, we get an $\arcsin$.
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