  According to Haar's Theorem, every compact group $G$ admits a unique
(regular, right and) left-invariant Borel probability measure $\mu_G$. Let the
Haar integral (of $G$) denote the functional $\int_G:\mathcal{C}(G)\ni f\mapsto
\int f\,d\mu_G$ integrating any continuous function $f:G\to\mathbb{R}$ with
respect to $\mu_G$. This generalizes, and recovers for the additive group
$G=[0;1)\mod 1$, the usual Riemann integral: computable (cmp. Weihrauch 2000,
Theorem 6.4.1), and of computational cost characterizing complexity class
#P$_1$ (cmp. Ko 1991, Theorem 5.32). We establish that in fact every computably
compact computable metric group renders the Haar integral computable: once
asserting computability using an elegant synthetic argument, exploiting
uniqueness in a computably compact space of probability measures; and once
presenting and analyzing an explicit, imperative algorithm based on 'maximum
packings' with rigorous error bounds and guaranteed convergence. Regarding
computational complexity, for the groups $\mathcal{SO}(3)$ and
$\mathcal{SU}(2)$ we reduce the Haar integral to and from Euclidean/Riemann
integration. In particular both also characterize #P$_1$. Implementation and
empirical evaluation using the iRRAM C++ library for exact real computation
confirms the (thus necessary) exponential runtime.
