There are several functions in base R for differentiation, integration, solving system of equations, etc. E.g. `solve`, `stats::D`, `stats::deriv`, `stats::integrate`, `stats::numericDeriv`.

R package Deriv for symbolic differentiation, it allows user to supply custom rules for differentiation.[1]

R package numDeriv[2] for calculating numerical approximations to derivatives.

R package gmp[3] and Rmpfr[4] provide multiple precision arithmetic and floating point operations. They also include some special functions, e.g. Rmpfr::integrateR for numerical integration.

R package mpc[5] available at R forge. It provides multiple precision arithmetic for complex numbers.

R package rSymPy[6] provides an interface to ‘SymPy’ library in python via rJava.

R package Ryacas[7] provides an interface to the ‘Yacas’ computer algebra system. It is easier to install compared to `rSymPy`.

R package symengine[8] is an R interface to the SymEngine C++ library for symbolic computation.

[1] https://cran.r-project.org/web/packages/Deriv/index.html

[2] https://cran.r-project.org/web/packages/numDeriv/index.html

[3] https://cran.r-project.org/web/packages/gmp/index.html

[4] https://cran.r-project.org/web/packages/Rmpfr/index.html

[5] http://mpc.r-forge.r-project.org/

[6] https://cran.r-project.org/web/packages/rSymPy/index.html

[7] https://cran.r-project.org/web/packages/Ryacas/index.html

[8] https://github.com/symengine/symengine.R

I use R much more often than Mathematica and think is great for many reasons, but there are places where Mathematica is on another level. My mathematical maturity isn't high enough to really get how it's done or describe it well, but Mathematica has a way of being shockingly consistent across concepts and has pretty thorough documentation that can even help you learn the topics. R is very inconsistent even in the internal library, and documentation quality runs from best around to worse than no documentation.

There are also little nifty things like for image processing you can have a hard coded image show up in your code (I like plain text better but it's cool and future-techy). Distributions (as in normal, binomial, Poisson, etc) are a type and PDFs and CDFs can be obtained from them consistently rather than having to remember the different parameters of dnorm, dbinorm, etc.

I would love a real Mathematica expert to give us more. That's the real drawback of the closed ecosystem there is so much less information about it out there, fewer code samples, etc.

Great information, thanks. I do totally agree about documentation for R libraries. It can be very hit and miss.