Table 6. Tested bonus equations.

Goldstein 8.56 is for the special case where the vectors p and A are parallel.

Source	Equation	Solved	Solved by Eureqa	Methods used
Rutherford scattering	$A={\left(\frac{Z_1{Z}_2\mathrm{\alpha }\mathrm{\hslash }c}{4E{\mathit{\text{sin}}}^2\left(\frac{\mathrm{\theta }}{2}\right)}\right)}^2$	Yes	No	da, bf-sqrt
Friedman equation	$H=\sqrt{\frac{8\mathrm{\pi }G}{3}\mathrm{\rho }-\frac{k_f{c}^2}{a_f^2}}$	Yes	No	da, bf-squared
Compton scattering	$U=\frac{E}{1+\frac{E}{m{c}^2}(1-\text{cos} \mathrm{\theta })}$	Yes	No	da, bf
Radiated gravitational wave power	$P=-\frac{32}{5}\frac{G^4}{c^5}\frac{{(m_1{m}_2)}^2(m_1+m_2)}{r^5}$	No	No	–
Relativistic aberration	${\mathrm{\theta }}_1=\text{arccos}\left(\frac{\text{cos} {\mathrm{\theta }}_2-\frac{v}{c}}{1-\frac{v}{c}\text{cos} {\mathrm{\theta }}_2}\right)$	Yes	No	da, bf-cos
N-slit diffraction	$I=I_0{\left[\frac{\text{sin}(\mathrm{\alpha }/2)}{\mathrm{\alpha }/2}\frac{\text{sin}(N\mathrm{\delta }/2)}{\text{sin}(\mathrm{\delta }/2)}\right]}^2$	Yes	No	da, sm, bf
Goldstein 3.16	$v=\sqrt{\frac{2}{m}(E-U-\frac{L^2}{2m{r}^2})}$	Yes	No	da, bf-squared
Goldstein 3.55	$k=\frac{m{k}_G}{L^2}(1+\sqrt{1+\frac{2E{L}^2}{m{k}_G^2}}\text{cos}({\mathrm{\theta }}_1-{\mathrm{\theta }}_2\left)\right)$	Yes	No	da, sym–, bf
Goldstein 3.64 (ellipse)	$r=\frac{d(1-{\mathrm{\alpha }}^2)}{1+\text{αcos}({\mathrm{\theta }}_1-{\mathrm{\theta }}_2)}$	Yes	No	da, sym–, bf
Goldstein 3.74 (Kepler)	$t=\frac{2\mathrm{\pi }{d}^{3/2}}{\sqrt{G(m_1+m_2)}}$	Yes	No	da, bf
Goldstein 3.99	$\mathrm{\alpha }=\sqrt{1+\frac{2{\mathrm{ϵ}}^{2}E{L}^2}{m{(Z_1{Z}_2{q}^2)}^2}}$	Yes	No	da, sym*, bf
Goldstein 8.56	$E=\sqrt{{(p-q{A}_{\text{vec}})}^2{c}^2+m^2{c}^4}+q{V}_e$	Yes	No	da, sep+, bf-squared
Goldstein 12.80	$E=\frac{1}{2m}[p^2+m^2{\mathrm{\omega }}^2{x}^2(1+\mathrm{\alpha }\frac{x}{y})]$	Yes	Yes	da, bf
Jackson 2.11	$F=\frac{q}{4\mathrm{\pi }\mathrm{ϵ}{y}^2}[4\mathrm{\pi }\mathrm{ϵ}{V}_{e}d-\frac{\mathit{qd}{y}^3}{{(y^2-d^2)}^2}]$	No	No	–
Jackson 3.45	$V_e=\frac{q}{{(r^2+d^2-2\mathit{dr}\text{cos} \mathrm{\alpha })}^{\frac{1}{2}}}$	Yes	No	da, bf-inv
Jackson 4.60	$V_e=E_f\text{cos} \mathrm{\theta }(\frac{\mathrm{\alpha }-1}{\mathrm{\alpha }+2}\frac{d^3}{r^2}-r)$	Yes	No	da, sep*, bf
Jackson 11.38 (Doppler)	${\mathrm{\omega }}_0=\frac{\sqrt{1-\frac{v^2}{c^2}}}{1+\frac{v}{c}\text{cos} \mathrm{\theta }}\mathrm{\omega }$	Yes	No	da, cos-input, bf
Weinberg 15.2.1	$\mathrm{\rho }=\frac{3}{8\mathrm{\pi }G}(\frac{c^2{k}_f}{a_f^2}+H^2)$	Yes	Yes	da, bf
Weinberg 15.2.2	$p_f=-\frac{1}{8\mathrm{\pi }G}[\frac{c^4{k}_f}{a_f^2}+c^2{H}^2(1-2\mathrm{\alpha })]$	Yes	Yes	da, bf
Schwarz 13.132 (Klein-Nishina)	$A=\frac{\mathrm{\pi }{\mathrm{\alpha }}^2{\mathrm{\hslash }}^2}{m^2{c}^2}{\left(\frac{{\mathrm{\omega }}_0}{\mathrm{\omega }}\right)}^2[\frac{{\mathrm{\omega }}_0}{\mathrm{\omega }}+\frac{\mathrm{\omega }}{{\mathrm{\omega }}_0}-{\text{sin}}^2\mathrm{\theta }]$	Yes	No	da, sym/, sep*, sin-input, bf