Redberry is an open source computer algebra system designed for algebraic manipulations with tensors. Redberry is a computer algebra system which considers both tensors and indexless expressions in a common way.

Key features:
• Programming language with internal support of symbolic algebra
• Tensor symmetries, multiple index types, dummy indices handling, $\LaTeX$-style I/O, mappings of tensor indices
• A wide range of tensor-specific transformations and simplification routines
• Tools for calculations in High Energy Physics: Feynman diagrams and one-loop counterterms
• Extensive API for developers

Example: one-loop counterterms
This code calcultes one-loop counterterms of the minimal second order operator: $D_i{}^j\,=\,\delta_i{}^j\,\Box + W_i{}^j$
import cc.redberry.groovy.Redberry import static cc.redberry.groovy.RedberryPhysics* import static cc.redberry.groovy.RedberryStatic.* /* * One-loop counterterms for minimal second order operator */ use(Redberry) { def iK = 'iK_a^b = d^b_a'.t def K = 'K^lm_a^b = d^b_a*g^{lm}'.t def S = 'S^lab = 0'.t def W = 'W_a^b = W_a^b'.t def F = 'F_lmab = F_lmab'.t def div = RedberryPhysics.oneloopdiv2(iK, K, S, W, F) def counterterms = EliminateDueSymmetries >> div.counterterms println counterterms }
Multiplying the produced result by $1\left/16\pi(d-4)\right.$ and integrating over the space-time volume gives: $\Gamma^{(1)}_{\infty} = \frac{1}{16\pi(d-4)} \int d^4 x \sqrt{-g} \left( \frac{1}{30} R^2 +\frac{1}{12} F_{\nu\beta }{}^{\epsilon}{}_{\rho} F^{\nu\beta\rho}{}{\epsilon }+\frac{1}{15} R_{\delta \nu } R^{\delta \nu }+\frac{1}{2} W^{\alpha }{}_{\rho} W^{\rho}_{\alpha}+\frac{1}{6} R W^{\beta }{}_{\beta}\right),$ where $F_{\mu\nu\alpha\beta}$ is a curvature tensor with respect to the principal bundle, $R_{\mu\nu}$ is a Ricci tensor and $R$ is a Riemann scalar curvature.