Let $B$ be a standard Brownian motion and $t>0$.
It can easily be shown that $$\int_0^t B(s) dB(s) = \frac{B^2(t) - t}{2}$$ using integration by parts formula.
Do we have such analytic form for $$\int_0^t B^n(s) dB(s)$$
for all $n \in \mathbb N$ ? Or at least for some $n$ ? Like $n=2$ or $n=3$ ?