Problem Key |
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\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 |
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\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} |
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\times |
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2\infty - 3 \rightarrow \infty |
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s 47.2 µs |
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\frac{e^x-1}{e^{x/2}+1} \rightarrow e^{x/2} - 1 |
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s 2.59 ms |
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A(x \geq y, y \geq z, z \geq x); x = z? |
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s 1.57 ms |
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A(x > y, y > 0); 2x^2 > 2y^2? |
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\frac{\cos(3x)}{\cos x} \rightarrow \cos^2 x - 3\sin^2 x |
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\frac{\cos(3x)}{\cos x} \rightarrow 2\cos(2x) - 1 |
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A(x,y > 0); x^{1/n}y^{1/n} - (xy)^{1/n} \rightarrow 0 |
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\log(\tan(\frac{1}{2}x + \frac{\pi}{4})) - \sinh^{-1}(\tan(x)) \rightarrow 0 |
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\log\frac{2\sqrt{r} + 1}{\sqrt{4r + 4\sqrt{r} + 1}} \rightarrow 0 |
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\frac{\sqrt{xy|z|^2}}{\sqrt{x}|z|} \rightarrow \frac{\sqrt{xy}}{\sqrt{x}} \not\rightarrow \sqrt{y} |
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s 2.11 ms |
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Note \sqrt{x} = \pm\sqrt{x} |
\frac{x=0}{2}+1 \rightarrow \frac{x}{2}+1=1 |
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S(e^{2x} + 2e^x + 1 = z,x) |
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s 4.85 ms |
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S((x+1)(\sin^2x + 1)^2\cos^3(3x)=0,x) |
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M^{-1}, where M = [[x,1],[y,e^z]] |
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s 3.93 ms |
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\sum_{k=1}^n k^3 \rightarrow \frac{n^2(n+1)^2}{4} |
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s 24.6 ms |
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\sum_{k=1}^\infty(\frac{1}{k^2} + \frac{1}{k^3}) \rightarrow \frac{\pi^2}{6} + \zeta(3) |
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\prod_{k=1}^nk \rightarrow n! |
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s 5.82 ms |
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\lim_{n\rightarrow\infty}(1 + \frac{1}{n})^n \rightarrow e |
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s 6.93 ms |
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\lim_{x\rightarrow 0}\frac{\sin x}{x} \rightarrow 1 |
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s 5.95 ms |
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\lim_{x\rightarrow 0}\frac{1-\cos x}{x^2} \rightarrow \frac{1}{2} |
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\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} |
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\frac{d}{dx}(\int\frac{1}{x^3+2}dx) \rightarrow \frac{1}{x^3+2} |
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\int\frac{1}{a+b\cos x}dx (a < b) |
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\frac{d}{dx}\int\frac{1}{a+b\cos x}dx = \frac{1}{a+b\cos x} |
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\frac{d}{dx}|x| \rightarrow \frac{x}{|x|} |
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\int|x|dx \rightarrow \frac{x|x|}{2} |
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\int\frac{x}{\sqrt{1+x}+\sqrt{1-x}}dx \rightarrow \frac{(1+x)^{3/2}+(1-x)^{3/2}}{3} |
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\int\frac{\sqrt{1+x}+\sqrt{1-x}}{2}dx \rightarrow \frac{(1+x)^{3/2}+(1-x)^{3/2}}{3} |
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\int_{-1}^1\frac{1}{x}dx \rightarrow 0 (p.v.) |
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\int_{-1}^1\frac{1}{x^2}dx \rightarrow (div) |
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\int_0^1\sqrt{x + \frac1x - 2}dx \rightarrow \frac43 |
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\int_1^2\sqrt{x + \frac1x - 2}dx \rightarrow \frac{4-\sqrt8}3 |
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\int_0^2\sqrt{x + \frac1x - 2}dx \rightarrow \frac{8-\sqrt8}3 |
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A(a>0); \int_{-\infty}^\infty\frac{\cos x}{x^2+a^2}dx \rightarrow \frac\pi ae^{-a} |
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A(0 < a < 1); \int_0^\infty\frac{t^{a-1}}{t+1}dt \rightarrow \frac{\pi}{\sin(\pi a)} |
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T(\frac1{\sqrt{1-(x/c)^2}},x=0) |
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T((\log x)^ae^{-bx},x=1) |
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T(\log(\sinh z) + \log(\cosh(z + w))) |
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T(\log(\frac{\sin x}{x}), x=0) |
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s 5.19 ms |
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at order 20 |