If systematically applied, the integration rules provided on this website can determine the
antiderivative of a wide variety of mathematical expressions.
As proof, a ** Ru**le-

To download *Rubi***4.15**, click on the **Download Rubi** menu option above.
This version expands the class of algebraic and other expressions ** Rubi** can integrate,
and often finds simpler, more concise antiderivatives in fewer steps than earlier versions.

- Int[
*expn*,*var*] returns the antiderivative (indefinite integral) of*expn*with respect to*var*. - Int[
*expn*,*var*, Step] displays the first step used to integrate*expn*with respect to*var*, and returns the intermediate result. - Int[
*expn*,*var*, Steps] displays all the steps used to integrate*expn*with respect to*var*, and returns the antiderivative. - Int[
*expn*,*var*, Stats], before returning the antiderivative of*expn*with respect to*var*, displays a list of statistics of the form {*a*,*b*,*c*,*d*,*e*} where *a*is the number of steps used to integrate*expn*,*b*is the number of distinct rules used to integrate*expn*,*c*is the leaf count size of*expn*,*d*is the leaf count size of the antiderivative of*expn*, and*e*is the rule-to-size ratio of the integration (i.e. the quotient of elements*b*and*c*).- Int[{
*expn1*,*expn2*, ...}, var] returns a list of the antiderivatives of*expn1*,*expn2*, ... each with respect to*var*. - Int[
*expn*, {*var*,*a*,*b*}] returns the limit of the antiderivative of*expn*as*var*approaches*b*minus the limit as*var*approaches*a*. Note that this difference will**not**always equal the definite integral of*expn*from*a*to*b*.

The last element of the list of statistics displayed by ** Rubi**'s Int[

I would like to challenge the community of ** Rubi** users to find the hardest problem the system can integrate.
To that end I will award $1000 (U.S.) to the person who sends me before August 1, 2018 the expression having the largest rule-to-size ratio
as displayed by the Int[

The hardest integral received thus far is Int[ArcCoth[x^16]^2,x] which has a rule-to-size ratio of 7.5. Please only send entries having ratios greater than 7.5.

** Rubi** dramatically out-performs

The following chart shows the percentage of optimal antiderivatives found by the three integrators for various types of integrands:

For example, it shows ** Mathematica** has the most difficulty with integrands involving trig functions; whereas

Integration Test Suite Results | |||||||||||||||

Maple 18 | Mathematica 10 | Rubi 4.11 | |||||||||||||

Integrand Types | Problems | Optimal | Nonoptimal | Fail | Optimal | Nonoptimal | Fail | Optimal | Nonoptimal | Fail | |||||

Algebraic linear functions | 4258 | 3533 | 392 | 333 | 3940 | 304 | 14 | 4253 | 5 | 0 | |||||

Algebraic quadratic functions | 5935 | 4591 | 1130 | 214 | 5626 | 283 | 26 | 5935 | 0 | 0 | |||||

Algebraic binomial functions | 6481 | 4985 | 874 | 622 | 5976 | 483 | 22 | 6475 | 6 | 0 | |||||

Algebraic trinomial functions | 2022 | 1447 | 451 | 124 | 1779 | 213 | 30 | 2021 | 1 | 0 | |||||

Miscellaneous algebraic functions | 1722 | 1185 | 344 | 193 | 1401 | 192 | 129 | 1708 | 7 | 7 | |||||

Exponential functions | 856 | 650 | 69 | 137 | 746 | 76 | 34 | 852 | 0 | 4 | |||||

Trig functions | 18050 | 8776 | 6826 | 2448 | 11878 | 5395 | 777 | 18026 | 14 | 10 | |||||

Hyperbolic functions | 3905 | 2098 | 1274 | 533 | 3240 | 626 | 39 | 3905 | 0 | 0 | |||||

Logarithm functions | 1058 | 410 | 153 | 495 | 909 | 117 | 32 | 1057 | 0 | 1 | |||||

Inverse trig functions | 3603 | 2567 | 407 | 629 | 3236 | 224 | 143 | 3599 | 2 | 2 | |||||

Inverse hyperbolic functions | 4877 | 2401 | 597 | 1879 | 4439 | 253 | 185 | 4872 | 1 | 4 | |||||

Special functions | 1239 | 644 | 59 | 536 | 991 | 29 | 219 | 1239 | 0 | 0 | |||||

Independent test problems | 1424 | 1146 | 173 | 105 | 1247 | 158 | 19 | 1373 | 28 | 23 | |||||

Totals | 55430 | 34433 | 12749 | 8248 | 45408 | 8353 | 1669 | 55315 | 64 | 51 | |||||

Percentages | 62.1% | 23.0% | 14.9% | 81.9% | 15.1% | 3.0% | 99.8% | 0.1% | 0.1% | ||||||

The following summarizes the meaning of the numbers under the column headings in the above table:

**Problems**: the number of integration problems for each type of integrand.**Optimal**: for, the number of results identical to the optimal antiderivative; for the other systems, the number of results no more than twice the size of the optimal antiderivative, based on leaf counts.*Rubi***Nonoptimal**: for, the number of results that are correct*Rubi**but*not identical to the optimal antiderivative; for the other systems, the number of results that are more than twice the size of the optimal antideriative.**Fail**: the number of problems the system returns unintegrated but that are actually integrable in terms of functions known by the system, plus the number of problems the system fails to integrate within a 120 second time limit.

The results of extensive, independently conducted, comparative testing of ** Rubi** and the built-in symbolic integrators of several computer algebra systems
is available at Computer Algebra Independent Integration Tests.

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