2.6.0

This release is about the integrator. #233
and #718 asked for "all of Rubi's rules,
but better", and about five hundred of this release's 706 pull requests move integration forward
in every Rubi chapter, from rational functions to the special functions. Each new route is checked
by differentiating its answer back. On Rubi's integration test-suite, 2.5.0 solves 7,580 of the
48,305 problems this library can state, and answers 81 of them wrongly. 2.6.0 solves all 48,305, and
answers none wrongly (the table under Integration). The rest of the release does three things. It makes statements
into data: quantifiers, congruences, divisibility, set relations, and closed forms of sums. It
rebuilds numerical evaluation on fixed-point series and interval arithmetic. And it moves the test
harnesses into CI. No tier of the Math OS roadmap
is completed by this release (see the note on #746).

Every changed answer is in BREAKING-CHANGES.md
under 2.6.0 — since 2.5.0, with the old value, the new one and why. Read it first if your code
depends on an answer. It holds 516 entries. 297 are integrals that now have an answer where 2.5.0
had none. Fourteen open as silent changes, where the call still succeeds and returns something
else. The one to read before any other is the first: | is divisibility now, and was
disjunction. "2 | 6" parsed as 2 or 6 and is now 2 divides 6. Writing or restores the old
reading everywhere, and it is what the library has always printed.

AssemblyVersion stays 2.0.0.0. The recorded public surface has 727 additions and no
removals, so code that compiled against 2.5.0 still compiles, and code bound against it still
binds. The assemblies now report the release in their file and informational versions. The 2.5.0
DLL reported 2.3.0 (#1679).

Wrong answers fixed

Each item has an entry in BREAKING-CHANGES.md with both values.

• A value substituted under a binder was captured by the name the binder binds. This affected
  an integral, a sum, a product, a derivative, a limit and a set builder
  (#1808, #1811).
• Simplify split a power of a quotient across the branch cut, so sqrt(x) * sqrt(1/x) was 1
  (#1734, #1755).
• A root of an even power was read as the power. (u^2)^(3/2) is |u|^3, not u^3
  (#1387, #1388, #1462, #1477). The same
  shape inside a quadratic is now read as the modulus of a linear
  (#1670, #1672).
  2.5.0 answered sec^(3/2)/(b sec)^(5/2) without the sign that makes it right where b and the
  secant are both negative. #1388 withdrew that answer.
• A fractional power of a product of trigonometric functions lost its sign (#1533).
• The rational canonical form dropped the points where the expression has no value
  (#1618, #1740).
• A power of a constant below 1e-50 was simplified to zero
  (#1769, #1772). The floor and the
  ceiling of a large rational went through a decimal and came out wrong
  (#1807, #1810).
• NaN was returned as the antiderivative of something that has one, five times over: 1/(a x^2)
  (#1257), an exponent read as written rather than as a number (#1267), a quadratic with a
  coefficient off the real line (#1598,
  #1599), an exponent that is -1 only as a value (#1723), and an exponential times a cosine with a
  complex rate, e^(-i x) cos(x) (#1380). 0 b^i was NaN where the power is defined
  (#1865, #1866), and sqrt(a) x
  differentiated to NaN at a = 0, because a constant was differentiated by the chain rule (#2014).
• Antiderivatives that were wrong on part of the line. sqrt(-x) (x + 1)/x was the negative of
  the right value for every x < 0: the long division read (c x)^r as c^r x^r for a negative c
  (#1957). 1/(asin(sqrt(1 + b x^2)) sqrt(1 + b x^2)) was wrong for every negative x (#1919), and
  1/asin(sin(x)) wherever cos(x) < 0 (#2048). (1/x) provided a > 0 integrated to ln(x/a),
  which has a value where a < 0 and the integrand has none (#2057).
• lim ln(x) + ln(x + 1) - ln(x^2 + x) as x → -∞ was 0, where it is 2 pi i
  (#2022, #2029).
• Evaluation and printing. A complex division by a number far from unit magnitude was 0
  (#1372, #1374). A constant and a cached
  evaluation kept the precision they were first evaluated at: pi at 300 digits after a 100-digit
  evaluation had 100 (#1367, #1368). A
  complex number made with the downcasting off printed as a different number once it was back on
  (#1611).
• Solve answered the empty set for a statement it could not read
  (#1409, #1440), and for an equation
  between a statement and a truth value (#1549, #1565).
  An undecided member of an intersection was dropped from it
  (#1680, #1681). An equation the solver
  cannot invert was answered with no roots (#1504).
• Two intervals that touched were joined before the empty one was checked
  (#1634, #1636).
• A piecewise claimed a domain that one of its cases did not have
  (#1817, #1819). A case whose condition
  is NaN is now passed over (#1764). A domain condition on a name that a binder binds now stays
  in the binder (#1632, #1633).
• Antiderivatives built through an even root now say where the root is real, where before they
  were silently wrong on the other side (#1431, #1532, #1594, #1663, #1667, #1669, #1671). Two
  related fixes: the third case of a binomial differential is now right for negative x (#1720),
  and so is the arcsecant below -1 (#1722).
• erfc, erfi, Ei, li, Si, Ci, Shi and Chi were read as products of a variable
  with their argument. They are functions now (#1506, #1526, #1529).
• Hundred-digit evaluation is now right to the last digit. sin(1), cos(1) and tan(1) were
  one to three units off in the last place, complex sines about twenty units off, and
  (3 ± 2i)^(3 ± 2i) up to 289 off. Nineteen pinned values moved, every one towards the true value
  (#1338, #1342, #1346, #1347).
• With downcasting off, a complex modulus close to a rational came back as that rational. The
  factorial far from zero was 24 orders of magnitude out
  (#1614, #1616, #1619).

Integration

Rubi's test-suite, the whole of it that AngouriMath can state: 48,305 problems whose integrand and
whose antiderivative use only functions this library has. Each was run at a 5-second budget, and every
answer was differentiated back and compared with the integrand numerically, never with Rubi's answer.

Family	Problems	2.5.0	2.6.0
0, independent suites	1,814	744	1,814
1, algebraic	20,179	4,640	20,179
2, exponentials	650	124	650
3, logarithms	1,382	141	1,382
4, trigonometric	14,107	1,479	14,107
5, inverse trigonometric	1,774	76	1,774
6, hyperbolic	3,690	338	3,690
7, inverse hyperbolic	4,289	38	4,289
8, special functions	420	0	420
all	48,305	7,580 (15.7%)	48,305 (100%)

• Wrong answers: 2.5.0 has 81, and 2.6.0 has none. 2.6.0 has no partial answer, no timeout and no
  unevaluated integral either.
• 2.6.0 was measured on master 784f69d5, on 2026-10-22. 2.5.0 (ab7a8bd6) was measured on the
  same rows on 2026-10-18. The harness has since changed how it judges an integrand that is complex on
  the whole real line, an answer with no value on part of the integrand's real domain, and a
  disagreement at a far point, and 2.5.0 was not re-run under those changes.
• Family 8 counts 2.5.0 as 0. 2.5.0 has none of erfc, erfi, Ei, li, Si, Ci, Shi and
  Chi, and reads erfc(x) as a variable erfc times x. It "solved" 273 rows of a different
  problem, which the harness verified against the same misreading.
• Loss against 2.5.0: none. The two found on the way, (x - 3 sqrt(x))^(2/3) (-3 + 2 sqrt(x))/sqrt(x)
  and sinh(a + b x)^3 sinh(c + d x)^3, were fixed in #1943 and #1942.

Other cross-family PRs since #1832: #1882, #1902, #1909, #1919, #1940, #1943, #1954, #1996, #2016, #2019,
#2020, #2024, #2025, #2034, #2045, #2056, #2057.

New general methods: the Hermite reduction in coefficient arithmetic with p-adic lifting, which
reduces a denominator of degree sixty in a second (#1327); the Rothstein–Trager resultant, in
real terms (#1304, #1309, #1329); the Risch–Norman ansatz over towers of exponentials and
logarithms (#1361); Liouville's ansatz for an exponential times a rational function (#1287,
#1296, #1297); Euler's substitutions (#1289, #1298, #1299); Chebyshev's three cases of
the binomial differential (#1275, #1288, #1721, #1792); Bioche's rules (#1337, #1350); and
square-root and cube-root ansätze over algebraic number fields (#1305, #1325, #1326, #1329). Since #1832: Osborn's rule, asking a hyperbolic integrand as the trigonometric one it
mirrors (#1961, #2042, #2053); and an ansatz for a power of a function of one exponential beside a
polynomial in x, W times a rational function of x and the exponential (#2059).

1. Algebraic functions

Rational functions with symbolic parameters, split into coprime blocks, over symbolic quadratics,
biquadratics and quartics in x^2. Radicals of linears and quadratics, Euler's substitution, the
binomial differential, pseudo-elliptic and nested roots. The headline case: a quotient that took
half a minute takes half a second (#1236), and a polynomial no longer takes exponential time to
integrate (#1250). On the family 1 sample, #1390 alone took 133 to 140 of 228.
Whole family: 4,640 of 20,179 in 2.5.0, all 20,179 in 2.6.0.
PRs: #1236, #1240, #1243, #1248–#1250, #1254, #1259, #1269, #1275, #1276, #1283, #1284, #1289,
#1292, #1295, #1298–#1300, #1302, #1304, #1305, #1309, #1313–#1318, #1325–#1327, #1329, #1333,
#1336, #1339, #1340, #1348, #1351–#1353, #1360, #1389–#1393, #1476, #1477, #1491, #1492, #1494,
#1542, #1551, #1576, #1583, #1595, #1602, #1637, #1683, #1693, #1697–#1707, #1712–#1715, #1720,
#1721, #1724, #1726–#1728, #1741, #1742, #1756–#1758, #1774, #1775, #1777, #1780–#1787, #1790,
#1792, #1794, #1795, #1797, #1799, #1800, #1803–#1806, #1809, #1827, #1830, #1831, #1833, #1835–#1837, #1839–#1845, #1847–#1849, #1854, #1860, #1862, #1863, #1869–#1873, #1875, #1876, #1881, #1884, #1886–#1901, #1903, #1904, #1907, #1913, #1915, #1916, #1923, #1929, #1932, #1933, #1937, #1939, #1948, #1953, #1957, #1960, #1962, #1968, #1974, #1976, #1979, #1985–#1988, #1990, #1992–#1995, #1997, #1999, #2000, #2007, #2012, #2017, #2023, #2027, #2033, #2035, #2036, #2038, #2039, #2041, #2046, #2050, #2058.

2. Exponentials

Anything rational in e^(k x), bases other than e, symbolic slopes, and an exponential times a
trigonometric function at the same frequency. Also the Gaussian and its moments, and exponentials of
quadratics in x, in 1/x and in a logarithm, integrated onto the error functions. Family 2 sample:
87 → 104 of 108 (#1362).
Whole family: 124 of 650 in 2.5.0, all 650 in 2.6.0.
PRs: #1247, #1262, #1278, #1287, #1296, #1331, #1362, #1434, #1472, #1512, #1518–#1525, #1641, #1733,
#1748, #1864, #1935, #1942, #1950, #1951, #1989, #2008, #2010, #2040.

3. Logarithms

Integration by parts now follows LIATE and descends on the total power of its transcendental
factors. Also added: a function of ln(x) alone, powers of a logarithm's two linears under
t = L1/L2, and A + B ln(c x^n) by the closed reduction. Family 3 sample: 148 → 185 → 206 of 225
(#1363, #1379).
Whole family: 141 of 1,382 in 2.5.0, all 1,382 in 2.6.0.
PRs: #1263, #1271, #1272, #1281, #1297, #1303, #1320, #1355, #1361, #1363, #1379, #1407, #1410, #1663,
#1667, #1729, #1752, #1910, #1955, #2005, #2009, #2011, #2013, #2037, #2047, #2052.

4. Trigonometric functions

sin^m cos^n for every product of the six functions; the tangent, half-angle and homogeneous
substitutions; secant and cosecant recurrences; a cos + b sin rotated into one cosine; and
product-to-sum. In the 4.3 rows, the a + i a tan family now has a closed treatment, beside powers
of the secant, under roots, and to symbolic powers. One 4.3 answer went from 3.5 million characters
to 7 thousand (#1809).
Whole family: 1,479 of 14,107 in 2.5.0, all 14,107 in 2.6.0.
PRs: #1238, #1245, #1264, #1266, #1268, #1270, #1273, #1274, #1277, #1282, #1286, #1293, #1294, #1307,
#1311, #1312, #1322–#1324, #1332, #1334, #1335, #1337, #1350, #1354, #1401, #1405, #1474, #1475, #1479,
#1481, #1483–#1485, #1493, #1584, #1585, #1593, #1596, #1717, #1719, #1730, #1731, #1735, #1737,
#1743, #1744, #1747, #1749, #1754, #1760–#1768, #1771, #1778, #1779, #1791, #1796, #1798, #1801,
#1802, #1812–#1816, #1818, #1820–#1822, #1824–#1826, #1828, #1829, #1832, #1846, #1850, #1851, #1853, #1855–#1859, #1861, #1867, #1878, #1880, #1912, #1922, #1924, #1927, #1930, #1958, #1959, #1964, #1967, #1970, #1975, #1977, #1980, #1984, #1991, #1998, #2001, #2002, #2026, #2031, #2049, #2051, #2055.

5. Inverse trigonometric functions

The arcsecant and the arccosecant have antiderivatives. The substitution that undoes an inverse
function is now used, including for a linear argument and for powers of a + b arcsin(c x). Family 5
sample: 126 → 174 of 257 (#1380).
Whole family: 76 of 1,774 in 2.5.0, all 1,774 in 2.6.0.
PRs: #1261, #1279, #1291, #1330, #1380, #1489, #1536, #1722, #1917, #1918, #1925, #1926, #1934, #1938, #1973, #2048.

6. Hyperbolic functions

The hyperbolic functions have antiderivatives. Also added: the hyperbolic half angle, u = tanh(y)
with symbolic coefficients, and powers of a + i a csch. Family 6 sample: 278 → 301 of 417 (#1377).
Whole family: 338 of 3,690 in 2.5.0, all 3,690 in 2.6.0.
PRs: #1247, #1357, #1377, #1463–#1467, #1470, #1739, #1753, #1823, #1905, #1911, #1936, #1956, #1961, #1971, #1981–#1983, #2018, #2021, #2042, #2053, #2054, #2059.

7. Inverse hyperbolic functions

asech, acsch and atanh with symbolic and linear arguments, and e^(k atanh) and e^(k acoth)
beside powers of a linear. Answers are given on the function's domain and say so. Family 7 sample:
77 → 150 of 270 (#1385).
Whole family: 38 of 4,289 in 2.5.0, all 4,289 in 2.6.0.
PRs: #1385, #1438, #1451, #1456, #1457, #1459, #1460, #1487, #1488, #1571, #1572, #1669, #1931, #1952, #1965, #1969, #1978, #2028, #2032.

8. Special functions

erf, erfc, erfi, Ei, li, Si, Ci, Shi and Chi are functions. Each is evaluated to
any precision and compiled in double precision. Each is reached as an answer, integrated by parts,
and used as a substitution.
Whole family: 0 of 420 in 2.5.0, all 420 in 2.6.0.
PRs: #1506, #1526, #1527, #1529, #1530, #1535–#1537, #1609, #1613, #1621, #1631, #1638, #1639,
#1642–#1645, #1647, #1708, #1732, #1906, #1908, #1947, #1963, #1972.

Definite integrals

A definite integral is now taken piece by piece across its antiderivative's breaks. It is the limit
at a bound where the antiderivative is undefined, and it diverges across a pole (#1509, #1515).

Statements, sets and number theory

New in this release:

• Quantifiers forall, exists, exists! over a mandatory set (#1421, #1499, #1500, #1564, #1586).
• Congruences a = b (mod n), solved to a residue class and met by the Chinese remainder theorem
  (#1419, #1424, #1630).
• Divisibility | (#1241); lcm, prime(n), valuation(n, p); PP, the primes; min(S);
  ZZ* and ZZ+ (#1413, #1452, #1453, #1579, #1580, #1590).
• Set relations and functions: subset, powerset, image, preimage, aleph cardinalities,
  integer ranges, and families of sets (#1432, #1435, #1442, #1445, #1557, #1574, #1577, #1591, #1592).
• Sums: binomial(n, k) and its identities; the Iverson bracket iverson(P); sum(f, x in S);
  the pattern operator ...; closed forms of polynomial-times-power, factorial and binomial sums;
  and statements about a sum up to n decided by induction (#1411, #1439, #1444, #1446, #1458,
  #1461, #1538, #1550, #1567, #1575, #1597, #1624).
• ProofRecording collects the steps by which a quantified statement was decided (#1454).
• π is pi and the script ℯ is e (#1608).

Numerical evaluation

The logarithm, the exponential, the sine, the cosine and the arctangent are now fixed-point series
on System.Numerics.BigInteger, with argument reduction by a table (#1346, #1347, #1364, #1366). The
rational search behind every decimal operation is decided in a double and a double-double first
(#1342, #1344, #1365). The integrator's sampled checks evaluate in intervals (#1497, #1621). Constants and
cached evaluations follow the precision in force, per flow (#1368, #1371, #1514). Complex division
far from unit magnitude uses Smith's method (#1374). With downcasting off, integration, limits and
comparisons run on exact numbers (#1516, #1517, #1531, #1611, #1619).

Speed

Measured as a pair: v2.5.0 beside the last kernel change, in one run on one machine
(version_performance_control.md, #2060).
Allocation, which is deterministic: the evaluation rows fall 95–99%, ParseHard 98.3%, SolveEasy
94.9%, SimplifyHard 52.4% and SolveHard 15.9%. Derivate is up 5.65%, in five steps the gate
could not see one at a time, and is recorded unattributed. SimplifyEasy is +0.5% and SolveEasyMedium
+1.5%.

The steps below were each measured against the commit before them, and each is answer-identical unless
noted. They compose: SolveEasy's six recorded steps predict −94.9%, and the pair measures −94.89%.

ParseHard	−98.4% allocation, 1,525 → 25.6 µs: the parser predicts in SLL first (#1528)
EvalPolynomialFresh	−97.8% allocation, 349 → 9.4 µs: the rational search, powers and series (#1342)
EvalTranscendentalFresh	3.25 → 1.32 ms (#1342); ln at 100 digits 18.0 → 8.4 µs (#1366)
SolveEasy	−42.1% allocation, 390 → 222 µs: the root of a negative real is exactly imaginary (#1379)
SolveHard	112 → 64 ms: equality and hash leave the caches out (#1345)
SimplifyHard	−5.2% allocation: operator nodes cache their flat operand list (#1426)
integration	a degree-sixty Hermite reduction in a second (#1327); a cancellation token honoured inside expansions and linear solves, so a corpus run went from 733 s to 282 s (#1746)
erfc(100)	over 20 s → 3 ms (#1609)

Tools

• amcli answers one command and exits, as the standalone CLI did. amcli info describes an
  expression, and the Terminal's .deb installs amcli with a man page and completion
  (#1654, #1661, #1662, #1668).
• The MCP server is amcli mcp. It lives in Sources/MCP and builds against this repository
  (#1677).
• The Terminal renders LaTeX in the browser (#1495) and is released with the packages under their
  version (#1496, #1541, #1688).
• AsSingleFraction (#1622); a square matrix's characteristic polynomial (#1625); Factorize
  in a subexpression (#1422); matrices, the floors, the special functions, the factorial and the
  totient compile (#1425, #1605, #1613, #1620).

Checked in CI

The boundary, root, corpus, property, simplification-sweep, rule-set, crash, canonical-form and
arm-order harnesses all run in CI now, and so do the documentation samples from the wiki and the
website (#1650–#1653, #1656). Every test class names its area (#1623).

API and behaviour changes

All of these are in BREAKING-CHANGES.md.
The ones that change how existing input parses:

• | is divisibility, and was disjunction (#1241, the maintainer's decision on
  #1212).
• Names that were variables or products are now functions, keywords or sets: forall, exists,
  exists!, binomial, iverson, erf, erfc, erfi, Ei, li, Si, Ci, Shi, Chi, lcm,
  prime, valuation, min, PP, subset, powerset, image, preimage, π, ℯ, ..., ZZ*, ZZ+,
  and mod after an equation.
• Methods that threw now return null where there is no answer: InvertNode, the statement
  solver, and an inversion with no written preimage (#1510, #1559, #1560). Complexity saturates
  at int.MaxValue (#1601).

---

Pull requests in this release

706 PRs, listed by number below. They are the 707 commits of v2.5.0..ec0d9633 less bcec8f40 (Simplify report.md, by Hadrian Tang). #1709 and its revert #1716 cancel out.

Workspace, process and documentation, with no library change: #1375, #1381, #1384, #1398, #1404, #1412, #1416, #1418, #1429, #1436, #1447, #1471, #1480, #1482, #1511, #1617, #1626, #1696, #1709, #1716, #1834, #1914, #1920, #1921, #1941, #1946, #1949, #2060.

1236 1238 1240 1241 1243 1245 1246 1247 1248 1249 1250 1254 1257 1259 1261 1262 1263 1264 1266 1267
1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1286 1287 1288
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1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1368 1371 1374
1375 1377 1379 1380 1381 1384 1385 1388 1389 1390 1391 1392 1393 1395 1396 1397 1398 1399 1401 1402
1403 1404 1405 1406 1407 1408 1410 1411 1412 1413 1416 1417 1418 1419 1421 1422 1423 1424 1425 1426
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1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1470 1471 1472
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1627 1630 1631 1633 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1647 1650 1651 1652 1653
1654 1656 1657 1661 1662 1663 1667 1668 1669 1671 1672 1677 1679 1681 1682 1683 1685 1686 1687 1688
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1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731
1732 1733 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752
1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1771 1772 1773 1774
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1798 1799 1800 1801 1802 1803 1804 1805 1806 1809 1810 1811 1812 1813 1814 1815 1816 1818 1819 1820
1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1839 1840 1841
1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862
1863 1864 1866 1867 1869 1870 1871 1872 1873 1875 1876 1878 1880 1881 1882 1884 1886 1887 1888 1889
1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909
1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929
1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1946 1947 1948 1949 1950 1951
1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971
1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
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