up
PDF letter size
PDF legal size

Computer Algebra Independent Integration Tests
Mathematica 11, Rubi 4.9.8, Maple 2016 and Mupad 7.0 (Matlab 2016a)

Nasser M. Abbasi

August 29, 2016 compiled on — Monday August 29, 2016 at 06:15 AM
This report gives the result of running the computer algebra independent integration problems maintained by Albert Rich on Mathematica 11 (64 bit) Maple 2016 (64 bit), Rubi 4.9.8 and Mupad 7.0 (Matlab 2016a) on windows 7, 64 bit OS. The PC used is an Intel i7-3930k running at 3.20 GHz with 16 GB memory.

Total number of problems processed at the time of the report is 58469 The number of problems with no known antiderivative is 2499 The following table summarizes the overall result at the time of this report. The normalized size values in the following table are with respect to the optimal antiderivative size. For example, if a system produces an antiderivative of size 120 (leaf count) for a specific problem, and the optimal size for this problem is 100, then the normalized size is taken as 1.2.










mean median mean
percentage not CPU CPU mean median normalized
System solved solved (sec) (sec) size size size
















Rubi 99.923 45 0.974 0.216 146.102 107 1.006








Mathematica 97.987 1177 3.177 0.181 4831.423 105 17.891








Maple 86.858 7684 0.183 0.025 22340.470 136 120.232








Mupad 53.555 27156 2.067 0.250 910.501 93 4.093









Table 1: Summary of final results. Not solving a problem with no known antiderivative is counted as passed test

Below is the histogram for the normalized size for each system. Due to few but very large outliers in data and to make the histogram easier to view, binning is restricted to sizes up to 12 times the optimal size. The bin size used is 1.

pict pict

pict pict

Below is the histogram for the CPU time used. Due to few but large outliers in data, and to make the histogram easier to view binning is restricted to results up to 20 second of CPU time. The bin size used is 1 second.

pict pict

pict pict





Max CPU (sec) Problem location






Rubi 110.6 Problem 101 in 4_Trig_functions/3_Tangent/342/



Mathematica 600.1 Problem 774 in 6_Hyperbolic_functions/7/



Maple 499.2 Problem 351 in 4_Trig_functions/1_Sine/131/



Mupad 652.9 Problem 70 in 4_Trig_functions/3_Tangent/342/




Table 2: Maximum CPU problem and the integral location for each system





Max leaf size (actual) Problem location






Rubi 2954 Problem 411 in timofeev/



Mathematica 3302004 Problem 183 in 4_Trig_functions/5_Secant/521/



Maple 43437450 Problem 105 in 4_Trig_functions/7_Miscellaneous/71/



Mupad 630455 Problem 76 in 4_Trig_functions/3_Tangent/342/




Table 3: Largest result size (leaf size) and the integral location for each system

Specialized integral tests
A.F. Timofeev PDF HTML [705]
Kevin Charlwood PDF HTML [50]
Vladimir Bondarenko PDF HTML [35]
Manuel Bronstein PDF HTML [13]
Michael Wester PDF HTML [8]
Waldek Hebisch PDF HTML [7]
Tom Apostol PDF HTML [175]
Tony Hearn PDF HTML [284]
David Jeffrey PDF HTML [9]
Joel Moses thesis PDF HTML [113]
Martin Welz PDF HTML [41]
1. Algebraic functions
1. Linear Products
(1.2) (a+  bx )m (c+ dx )n  PDF HTML [1340]
(1.3)        m        n        p
(a+  bx ) (c+ dx )(e + fx)  PDF HTML [3033]
(1.4) (a+  bx )m (c+ dx )n(e + fx)p(g + hx )q  PDF HTML [77]
2. Quadratic Products
(2.2) (d+  ex )m (a+ bx + cx2)p  PDF HTML [3046]
(2.3) (d+  ex )m (f + gx)n(a+ bx + cx2)p  PDF HTML [3068]
(2.4) (a+  bx + cx2)p(d+ ex + fx2)q  PDF HTML [70]
(2.5) (g + hx)m(a + bx+ cx2)p(d+  ex + fx2 )q  PDF HTML [177]
3. Binomial Products
(3.2) (cx)m(a + bxn)p  PDF HTML [3061]
(3.3)        m         n           2 p
(d+  ex ) (f + gx) (a+ bx + cx )  PDF HTML [409]
(3.4) (ex)m(a + bxn)p(c+ dxn)q  PDF HTML [1614]
(3.6) (gx)m(a + bxn)p(c+ dxn)q(e+ f xn)r  PDF HTML [167]
(3.7) (cx)mP q(x)(a+ bxn)p  PDF HTML [549]
(3.9) (cx)m(axj + bxn)p  PDF HTML [452]
(3.10)     m    j    k p      n q
(ex) (ax  + bx ) (c+ dx )  PDF HTML [299]
4. Trinomial Products
(4.2) (dx)m (a + bxn + cx2n)p  PDF HTML [1622]
(4.4) (fx)m (d + exn)q(a+ bxn + cx2n)p  PDF HTML [613]
(4.5)     m             n    2n p
(dx)  Pq(x)(a+ bx  + cx  )  PDF HTML [167]
(4.7) (dx)m (axq + bxn + cx2n−q)p  PDF HTML [122]
5. Miscellaneous
(5.1) Rational functions PDF HTML [269]
(5.2) Algebraic functions PDF HTML [679]
(5.3) Expansion problems PDF HTML [109]
(5.4) Substitution problems PDF HTML [364]
(5.5) Piecewise constant extraction PDF HTML [66]
2. Exponentials PDF HTML [856]
3. Logarithms
u(a+ blog(c(d(e+ fx)p)q))n  PDF HTML [251]
Other Logarithms PDF HTML [819]
4. Trig functions
1. Sin
1.0.1 (bsin)n  PDF HTML [62]
1.0.2 (bsin)m(dcos)n  PDF HTML [425]
1.1.1 (a+  bsin)n  PDF HTML [93]
1.1.2       p         m
(ccos) (a + bsin)  PDF HTML [651]
1.2.1 (a+  bsin)m (c + dsin)n  PDF HTML [864]
1.2.2       p         m         n
(gcos) (a+ b sin) (c + dsin)  PDF HTML [1478]
1.2.3 (gsin)p(a + bsin)m(c+ d sin)n  PDF HTML [51]
1.3.1 (a+  bsin)m (c + dsin)n(A + B sin)  PDF HTML [356]
1.4.1          m                 2
(a+  bsin)  (A  + B sin +C sin )  PDF HTML [5]
1.4.2 (a+  bsin)m (c + dsin)n(A + B sin +C sin2)  PDF HTML [36]
1.8           m         n
(a + bsin) (c+ dtrig)  PDF HTML [9]
1.9 trigm(a + bsinn +c sin2n)p  PDF HTML [258]
2. Cos
2.0.1 (bcos)n  PDF HTML [259]
2.1.1 (a+  bcos)n  PDF HTML [56]
2.1.2 (csin)p(a + bcos)m  PDF HTML [32]
2.2.1 (a+  bcos)m (c+ d cos)n  PDF HTML [933]
2.2.3 (gcos)p(a+ b cos)m (c + dcos)n  PDF HTML [1]
2.3.1          m          n
(a+  bcos) (c+ d cos) (A + B cos)  PDF HTML [628]
2.4.1 (a+  bcos)m (A + B cos+C  cos2)  PDF HTML [391]
2.4.2          m          n                 2
(a+  bcos) (c+ d cos) (A + B cos+C  cos )  PDF HTML [1535]
2.5 (e tan )m (a+ b cos)n  PDF HTML [20]
2.8           m          n
(a + bcos) (c+  dtrig)  PDF HTML [21]
2.9 trigm(a + bcosn+c cos2n)p  PDF HTML [92]
3. Tangent
3.0.1 (btan)n  PDF HTML [53]
3.0.2       m      n
(bsec) (dtan)  PDF HTML [86]
3.0.3 (bsin)m(dtan)n  PDF HTML [113]
3.0.4 (bcot)m(dtan )n  PDF HTML [35]
3.1.1 (a+  btan)n  PDF HTML [2]
3.2.1 (a+  btan)m(c+ d tan)n  PDF HTML [1345]
3.3.1           m          n
(a+  btan) (c+ d tan) (A + B tan)  PDF HTML [859]
3.4.2 (a+  btan)m(c+ d tan)n(A + B tan+C  tan2 )  PDF HTML [133]
3.5a        m          n
a (e sin)  (a + btan)  PDF HTML [93]
3.5b b(e sec)m (a + btan)n  PDF HTML [517]
39 trigm (a + btann +c tan2n )p  PDF HTML [209]
4. CoTangent
4.0.1       n
(bcot)  PDF HTML [38]
4.0.2 (bcsc)m(dcot)n  PDF HTML [11]
4.0.3       m      n
(bcos) (d cot)  PDF HTML [2]
4.1.1 (a+  bcot)n  PDF HTML [2]
4.2.1 (a+  bcot)m (c+ d cot)n  PDF HTML [104]
4.5a       m          n
(e cos)  (a+ bcot)  PDF HTML [19]
4.5b (e csc)m (a+ bcot)n  PDF HTML [21]
4.9     m         n      2np
trig (a + bcot +c cot  )  PDF HTML [94]
5. Secant
5.0.1 (bsec)n  PDF HTML [222]
5.1.2 (a+  bsec)m (dsec)n  PDF HTML [894]
5.2.1          m          n
(a+  bsec)  (c+ dsec)  PDF HTML [253]
5.2.3 (gsec)p(a+ bsec)m(c + dsec)n  PDF HTML [286]
5.3.1          m       n
(a+  bsec)  (dsec) (A +  B sec)  PDF HTML [646]
5.4.2 (a+  bsec)m (dsec)n(A +  B sec +C sec2)  PDF HTML [1458]
5.5 (e sin)m(a + bsec)n  PDF HTML [298]
5.6 (e tan )m (a+ b sec)n  PDF HTML [350]
5.7 trigm (a + bsecn+c sec(2n))p  PDF HTML [34]
6. Cosecant
6.0.1       n
(bcsc)  PDF HTML [69]
6.1.1 (a+  bcsc)n  PDF HTML [26]
6.1.2 (a+  bcsc)m (dcsc)n  PDF HTML [34]
6.3.1 (a+  bcsc)m (dcsc)n(A +  B csc)  PDF HTML [24]
6.4.2 (a+  bcsc)m (dcsc)n(A +  B csc +C csc2)  PDF HTML [1]
6.5 (e cos)m (a+ bcsc)n  PDF HTML [16]
6.6       m          n
(e cot)  (a+ bcsc)  PDF HTML [23]
6.7 trigm(a + bcscn+c csc2n)p  PDF HTML [27]
7. Miscellaneous
7.1 (ctrig)m (dtrig)n  PDF HTML [250]
7.2     m             n
trig (atrig + btrig)  PDF HTML [293]
7.3 (c + dx)mtrigntrigp  PDF HTML [633]
7.4 xm (a + btrign)p  PDF HTML [124]
7.5   m          n p
x  trig(a + bx )  PDF HTML [207]
7.6 (d + ex)mtrig(a + bx + cx2)n  PDF HTML [80]
7.7   m               n p
x  trig(a + blog(cx ))  PDF HTML [268]
7.8 f (a + bx + cx2)trig(d+ ex + fx2)n  PDF HTML [132]
7.9 Trig functions PDF HTML [882]
5. Inverse Trig functions
1. Inverse Sin
1.1 (f x)m(d+ ex2 )p(a + barcsin(cx)n  PDF HTML [723]
1.2 more inverse sin  functions PDF HTML [454]
2. Inverse Cosine PDF HTML [159]
3. Inverse Tangent PDF HTML [1829]
4. Inverse Cotangent PDF HTML [196]
5. Inverse Secant PDF HTML [166]
6. Inverse Cosecant PDF HTML [872]
6. Hyperbolic functions
1. Hyperbolic sine functions PDF HTML [893]
2. Hyperbolic cosine functions PDF HTML [591]
3. Hyperbolic tangent functions PDF HTML [477]
4. Hyperbolic cotangent functions PDF HTML [243]
5. Hyperbolic secant functions PDF HTML [412]
6. Hyperbolic cosecant functions PDF HTML [212]
7. Other Hyperbolic functions PDF HTML [1055]
7. Inverse Hyperbolic functions
1. Inverse Hyperbolic sine
1.1 (f x)m(d+ ex2 )p(a + barcsinh(cx)n  PDF HTML [647]
1.2 more inverse hyperbolic sine PDF HTML [369]
2. Inverse Hyperbolic cosine
2.1      m       2p                n
(f x) (d+ ex  )(a + barccosh(cx )  PDF HTML [561]
2.2 more inverse hyperbolic cosine functions PDF HTML [417]
3. Inverse Hyperbolic tangent PDF HTML [2111]
4. Inverse Hyperbolic cotangent PDF HTML [1186]
5. Inverse Hyperbolic secant PDF HTML [118]
6. Inverse Hyperbolic cosecant PDF HTML [63]
8. Special functions
1. Error functions PDF HTML [115]
2. Fresnel integral functions PDF HTML [126]
3. Exponential integral functions PDF HTML [149]
4. Trig integral functions PDF HTML [122]
5. Hyperbolic integral functions PDF HTML [122]
6. Gamma functions PDF HTML [168]
7. Zeta function PDF HTML [14]
8. Polylogarithm function PDF HTML [27]
9. Product logarithm function PDF HTML [396]