WesterBenchmarks

Problem Key

→

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

×

wrong answer/cannot do the problem

s sec/ms/μs

performs correctly in time s

>s sec/ms/μs

does not finish in time s

>.<,s or >.<,×

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.)

√2√3+4→1+√3 

×

2∞−3→∞

s 47.2 µs

ex−1ex/2+1→ex/2−1

s 2.59 ms

A(x≥y,y≥z,z≥x);x=z?

s 1.57 ms

A(x>y,y>0);2x2>2y2?

cosxcos(3x)→cos2x−3sin2x

cosxcos(3x)→2cos(2x)−1

A(x,y>0);x1/ny1/n−(xy)1/n→0

log(tan(21x+4π))−sinh−1(tan(x))→0

log2√r+1√4r+4√r+1→0

√x|z|√xy|z|2→√x√xy/→√y

Note √x=±√x

2x=0+1→2x+1=1

S(e2x+2ex+1=z,x)

s 4.85 ms

S((x+1)(sin2x+1)2cos3(3x)=0,x)

M−1, where M=[[x,1],[y,ez]]

s 3.93 ms

∑nk=1k3→4n2(n+1)2

s 24.6 ms

∑∞k=1(1k2+1k3)→6π2+ζ(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

\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