<s>[INST] How can I integrate $\int \frac{u^3}{(u^2+1)^3}du?$ How to integrate following
$$\int\frac{u^3}{(u^2+1)^3}du\,?$$
What I did is here:
Used partial fractions
$$\dfrac{u^3}{(u^2+1)^3}=\dfrac{Au+B}{(u^2+1)}+\dfrac{Cu+D}{(u^2+1)^2}+\dfrac{Au+B}{(u^2+1)^3}$$
After solving I got
$A=0, B=0, C=1, D=0, E=-1, F=0$
$$\dfrac{u^3}{(u^2+1)^3}=\dfrac{u}{(u^2+1)^2}-\dfrac{u}{(u^2+1)^3}$$
Substitute $u^2+1=t$, $2u\ du=dt$, $u\ du=dt/2$
$$\int\frac{u^3}{(u^2+1)^3}du=\int \frac{dt/2}{t^2}-\int \frac{dt/2}{t^3}$$
$$=\frac12\dfrac{-1}{t}-\frac{1}{2}\dfrac{-1}{2t^2}$$
$$=-\dfrac{1}{2t}+\dfrac{1}{4t^2}$$
$$=-\dfrac{1}{2(u^2+1)}+\dfrac{1}{4(u^2+1)^2}+c$$
My question: Can I integrate this with suitable substitution?  Thank you
 [/INST] One alternate method is thus: make the substitution $t=u^2$ which gives $dt = 2u\; du \iff du = dt/2 \sqrt t$. Then
$$\mathcal I := \int \frac{u^3}{(u^2+1)^3} du = \int \frac{t \sqrt t}{(t+1)^3} \cdot \frac{dt}{2 \sqrt t} = \frac 1 2\int\frac{t}{(t+1)^3}dt$$
Now let $w = t+1 \implies dw = dt$. Then
$$\mathcal I = \frac 1 2 \int \frac{w-1}{w^3}dw = \frac 1 2 \left( \int \frac 1 {w^2} dw - \int\frac{1}{w^3} dw \right)$$
I imagine you can finish things up from here.
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