Problem Key \rightarrow simplify A(...) assume ... S(...,x) solve ... for x T(...,x=b) Taylor series of ... based at b (p.v.) principal value (div) divergent
 Performance Key \times wrong answer/cannot do the problem s\ sec/ms/\mu s performs correctly in time s > s\ sec/ms/\mu s does not finish in time s >.<,s or >.<,\times very difficult to convince system to do what you want (regardless of performance)
 Problem Maple Mathematica GiNaC Maxima Sage Symbolics Notes (such as code used/version etc.) \sqrt{2\sqrt{3} + 4} \rightarrow 1 + \sqrt{3} \times 2\infty - 3 \rightarrow \infty s 47.2 µs \frac{e^x-1}{e^{x/2}+1} \rightarrow e^{x/2} - 1 s 2.59 ms A(x \geq y, y \geq z, z \geq x); x = z? s 1.57 ms A(x > y, y > 0); 2x^2 > 2y^2? \frac{\cos(3x)}{\cos x} \rightarrow \cos^2 x - 3\sin^2 x \frac{\cos(3x)}{\cos x} \rightarrow 2\cos(2x) - 1 A(x,y > 0); x^{1/n}y^{1/n} - (xy)^{1/n} \rightarrow 0 \log(\tan(\frac{1}{2}x + \frac{\pi}{4})) - \sinh^{-1}(\tan(x)) \rightarrow 0 \log\frac{2\sqrt{r} + 1}{\sqrt{4r + 4\sqrt{r} + 1}} \rightarrow 0 \frac{\sqrt{xy|z|^2}}{\sqrt{x}|z|} \rightarrow \frac{\sqrt{xy}}{\sqrt{x}} \not\rightarrow \sqrt{y} s 2.11 ms Note \sqrt{x} = \pm\sqrt{x} \frac{x=0}{2}+1 \rightarrow \frac{x}{2}+1=1 S(e^{2x} + 2e^x + 1 = z,x) s 4.85 ms S((x+1)(\sin^2x + 1)^2\cos^3(3x)=0,x) M^{-1}, where M = [[x,1],[y,e^z]] s 3.93 ms \sum_{k=1}^n k^3 \rightarrow \frac{n^2(n+1)^2}{4} s 24.6 ms \sum_{k=1}^\infty(\frac{1}{k^2} + \frac{1}{k^3}) \rightarrow \frac{\pi^2}{6} + \zeta(3) \prod_{k=1}^nk \rightarrow n! s 5.82 ms \lim_{n\rightarrow\infty}(1 + \frac{1}{n})^n \rightarrow e s 6.93 ms \lim_{x\rightarrow 0}\frac{\sin x}{x} \rightarrow 1 s 5.95 ms \lim_{x\rightarrow 0}\frac{1-\cos x}{x^2} \rightarrow \frac{1}{2} \frac{d^2}{dx^2}y(x(t)) \rightarrow \frac{d^2y}{dx^2}(\frac{dx}{dt})^2 + \frac{dy}{dx}\frac{d^2x}{dt^2} \frac{d}{dx}(\int\frac{1}{x^3+2}dx) \rightarrow \frac{1}{x^3+2} \int\frac{1}{a+b\cos x}dx (a < b) \frac{d}{dx}\int\frac{1}{a+b\cos x}dx = \frac{1}{a+b\cos x} \frac{d}{dx}|x| \rightarrow \frac{x}{|x|} \int|x|dx \rightarrow \frac{x|x|}{2} \int\frac{x}{\sqrt{1+x}+\sqrt{1-x}}dx \rightarrow \frac{(1+x)^{3/2}+(1-x)^{3/2}}{3} \int\frac{\sqrt{1+x}+\sqrt{1-x}}{2}dx \rightarrow \frac{(1+x)^{3/2}+(1-x)^{3/2}}{3} \int_{-1}^1\frac{1}{x}dx \rightarrow 0 (p.v.) \int_{-1}^1\frac{1}{x^2}dx \rightarrow (div) \int_0^1\sqrt{x + \frac1x - 2}dx \rightarrow \frac43 \int_1^2\sqrt{x + \frac1x - 2}dx \rightarrow \frac{4-\sqrt8}3 \int_0^2\sqrt{x + \frac1x - 2}dx \rightarrow \frac{8-\sqrt8}3 A(a>0); \int_{-\infty}^\infty\frac{\cos x}{x^2+a^2}dx \rightarrow \frac\pi ae^{-a} A(0 < a < 1); \int_0^\infty\frac{t^{a-1}}{t+1}dt \rightarrow \frac{\pi}{\sin(\pi a)} T(\frac1{\sqrt{1-(x/c)^2}},x=0) T((\log x)^ae^{-bx},x=1) T(\log(\sinh z) + \log(\cosh(z + w))) T(\log(\frac{\sin x}{x}), x=0) s 5.19 ms at order 20

WesterBenchmarks (last edited 2017-09-01 06:55:22 by chapoton)