It's common knowledge that complex analysis is helpful in computing a bunch of exact real integrals. Is there any occurence of quaternions/quaternion formalism helping in the same way? If not, what could be some plausible reasons?
Motivation
To address comments asking for motivation, I will try to outline some correspondences in quaternion analysis to complex analysis, especially those that in complex analysis are useful for evaluating definite real integrals.
Describing quaternions as $q= t+ix+jy+kz$, A. Sudberry's 1977 Quaternionic Analysis notes that if the Cauchy-Riemann analogue of $$\frac{\partial f}{\partial t}+i\frac{\partial f}{\partial x}+j\frac{\partial f}{\partial y}+k\frac{\partial f}{\partial z}=0$$ holds, then an analogue of Cauchy's theorem holds for $C$ a smooth, closed 3-manifold in $\mathbb{H}$:
$$\int_{C} f(q) \; Dq=0, \\ \text{for }Dq = (dx \,dy\, dz-i\,dt\, dy \, dz - j\,dt\, dx \, dz - k\,dt \, dx \, dy).$$
He also notes that for such regular functions, an analogue of Cauchy's integral formula holds:
$$f(q_0) = \frac{1}{2 \pi^2} \int_{\partial D}\frac{1}{|q-q_0|^2} (q-q_0)^{-1} Dq \,f(q) $$
for $D$ a domain in which $f$ is regular.
These results for integrals at least partially mirror those in complex analysis. In complex analysis, Cauchy's theorem and integral formula are very useful for deriving the results of definite real integrals. The higher dimensionality of the quaternionic integrals maybe suggests that real integrals of functions of three variables would be useful to consider, but I do not know. Can these results for quaternionic integrals of regular functions be used to evaluate definite real integrals?