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Jul 31, 2025
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Format code with JuliaFormatter v2.1.5 #114

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20 changes: 13 additions & 7 deletions src/homotopy.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -132,8 +132,10 @@ partial_int_rules = [
@rule 𝛷(~x, sech(~u)) => (atan(sinh(~u)), ~u)
@rule 𝛷(~x, coth(~u)) => (log(sinh(~u)), ~u)
# 1/trigonometric functions
@rule 𝛷(~x, 1 / sin(~u)) => (log(csc(~u) + cot(~u)) + log(sin(~u)), ~u)
@rule 𝛷(~x, 1 / cos(~u)) => (log(sec(~u) + tan(~u)) + log(cos(~u)), ~u)
@rule 𝛷(~x, 1 /
sin(~u)) => (log(csc(~u) + cot(~u)) + log(sin(~u)), ~u)
@rule 𝛷(~x, 1 /
cos(~u)) => (log(sec(~u) + tan(~u)) + log(cos(~u)), ~u)
@rule 𝛷(~x, 1 / tan(~u)) => (log(sin(~u)) + log(tan(~u)), ~u)
@rule 𝛷(~x, 1 / csc(~u)) => (cos(~u) + log(csc(~u)), ~u)
@rule 𝛷(~x, 1 / sec(~u)) => (sin(~u) + log(sec(~u)), ~u)
Expand All @@ -160,7 +162,8 @@ partial_int_rules = [
@rule 𝛷(~x, asech(~u)) => (asech(~u), ~u)
@rule 𝛷(~x, acoth(~u)) => (~u * acot(~u) + log(~u + 1), ~u)
# logarithmic and exponential functions
@rule 𝛷(~x, log(~u)) => (
@rule 𝛷(~x,
log(~u)) => (
~u + ~u * log(~u) +
sum(pow_minus_rule(~u, ~x, -1); init = one(~u)),
~u)
Expand All @@ -172,16 +175,19 @@ partial_int_rules = [
sum(sqrt_rule(~u, ~x, ~k); init = one(~u)), ~u)
@rule 𝛷(~x, sqrt(~u)) => (
sum(sqrt_rule(~u, ~x, 0.5); init = one(~u)), ~u)
@rule 𝛷(~x, 1 / sqrt(~u)) => (
@rule 𝛷(~x, 1 /
sqrt(~u)) => (
sum(sqrt_rule(~u, ~x, -0.5); init = one(~u)), ~u)
# rational functions
@rule 𝛷(~x, 1 / ^(~u::is_univar_poly, ~k::is_pos_int)) => (
@rule 𝛷(~x,
1 / ^(~u::is_univar_poly, ~k::is_pos_int)) => (
sum(pow_minus_rule(~u,
~x,
-~k);
init = one(~u)),
~u)
@rule 𝛷(~x, 1 / ~u::is_univar_poly) => (
@rule 𝛷(
~x, 1 / ~u::is_univar_poly) => (
sum(pow_minus_rule(~u, ~x, -1); init = one(~u)),
~u)
@rule 𝛷(~x, ^(~u, -1)) => (log(~u) + ~u * log(~u), ~u)
Expand Down Expand Up @@ -239,7 +245,7 @@ function pow_minus_rule(p, x, k; abstol = 1e-8)
end

function sqrt_rule(p, x, k)
h = Any[p^k, p^(k + 1)]
h = Any[p ^ k, p ^ (k + 1)]

Δ = diff(p, x)
push!(h, log(Δ / 2 + sqrt(p)))
Expand Down
10 changes: 8 additions & 2 deletions src/integral.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -99,7 +99,8 @@ function integrate(eq, x = nothing;

plan = NumericalPlan(abstol, radius, complex_plane, opt)

s, u, ε = integrate_sum(eq, x; plan, bypass, num_trials, num_steps,
s, u,
ε = integrate_sum(eq, x; plan, bypass, num_trials, num_steps,
show_basis, symbolic, max_basis, verbose, use_optim)

s = beautify(s)
Expand Down Expand Up @@ -206,7 +207,12 @@ end
# in a future version
function integrate_term(eq, x; kwargs...)
args = Dict(kwargs)
plan, num_steps, num_trials, show_basis, symbolic, verbose, max_basis = args[:plan],
plan, num_steps,
num_trials,
show_basis,
symbolic,
verbose,
max_basis = args[:plan],
args[:num_steps], args[:num_trials], args[:show_basis],
args[:symbolic], args[:verbose], args[:max_basis]

Expand Down
63 changes: 37 additions & 26 deletions src/rules.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -109,22 +109,28 @@ trigs_rules = [@rule tan(~x) => sin(~x) / cos(~x)
@rule sec(~x) => one(~x) / cos(~x)
@rule csc(~x) => one(~x) / sin(~x)
@rule cot(~x) => cos(~x) / sin(~x)
@rule sin(~n::is_int_gt_one * ~x) => sin((~n - 1) * ~x) * cos(~x) +
cos((~n - 1) * ~x) * sin(~x)
@rule cos(~n::is_int_gt_one * ~x) => cos((~n - 1) * ~x) * cos(~x) -
sin((~n - 1) * ~x) * sin(~x)
@rule tan(~n::is_int_gt_one * ~x) => (tan((~n - 1) * ~x) + tan(~x)) /
(1 - tan((~n - 1) * ~x) * tan(~x))
@rule csc(~n::is_int_gt_one * ~x) => sec((~n - 1) * ~x) * sec(~x) *
csc((~n - 1) * ~x) * csc(~x) /
(sec((~n - 1) * ~x) * csc(~x) +
csc((~n - 1) * ~x) * sec(~x))
@rule sec(~n::is_int_gt_one * ~x) => sec((~n - 1) * ~x) * sec(~x) *
csc((~n - 1) * ~x) * csc(~x) /
(csc((~n - 1) * ~x) * csc(~x) -
sec((~n - 1) * ~x) * sec(~x))
@rule cot(~n::is_int_gt_one * ~x) => (cot((~n - 1) * ~x) * cot(~x) - 1) /
(cot((~n - 1) * ~x) + cot(~x))
@rule sin(~n::is_int_gt_one *
~x) => sin((~n - 1) * ~x) * cos(~x) +
cos((~n - 1) * ~x) * sin(~x)
@rule cos(~n::is_int_gt_one *
~x) => cos((~n - 1) * ~x) * cos(~x) -
sin((~n - 1) * ~x) * sin(~x)
@rule tan(~n::is_int_gt_one *
~x) => (tan((~n - 1) * ~x) + tan(~x)) /
(1 - tan((~n - 1) * ~x) * tan(~x))
@rule csc(~n::is_int_gt_one *
~x) => sec((~n - 1) * ~x) * sec(~x) *
csc((~n - 1) * ~x) * csc(~x) /
(sec((~n - 1) * ~x) * csc(~x) +
csc((~n - 1) * ~x) * sec(~x))
@rule sec(~n::is_int_gt_one *
~x) => sec((~n - 1) * ~x) * sec(~x) *
csc((~n - 1) * ~x) * csc(~x) /
(csc((~n - 1) * ~x) * csc(~x) -
sec((~n - 1) * ~x) * sec(~x))
@rule cot(~n::is_int_gt_one *
~x) => (cot((~n - 1) * ~x) * cot(~x) - 1) /
(cot((~n - 1) * ~x) + cot(~x))
@rule 1 / sin(~x) => csc(~x)
@rule 1 / cos(~x) => sec(~x)
@rule 1 / tan(~x) => cot(~x)
Expand All @@ -140,18 +146,22 @@ trigs_rules = [@rule tan(~x) => sin(~x) / cos(~x)
@rule sin(~x + ~y) => sin(~x) * cos(~y) + cos(~x) * sin(~y)
@rule cos(~x + ~y) => cos(~x) * cos(~y) - sin(~x) * sin(~y)
@rule tan(~x + ~y) => (tan(~x) + tan(~y)) / (1 - tan(~x) * tan(~y))
@rule csc(~x + ~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(sec(~x) * csc(~y) + csc(~x) * sec(~y))
@rule sec(~x + ~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(csc(~x) * csc(~y) - sec(~x) * sec(~y))
@rule csc(~x +
~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(sec(~x) * csc(~y) + csc(~x) * sec(~y))
@rule sec(~x +
~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(csc(~x) * csc(~y) - sec(~x) * sec(~y))
@rule cot(~x + ~y) => (cot(~x) * cot(~y) - 1) / (cot(~x) + cot(~y))
@rule sin(~x - ~y) => sin(~x) * cos(~y) - cos(~x) * sin(~y)
@rule cos(~x - ~y) => cos(~x) * cos(~y) + sin(~x) * sin(~y)
@rule tan(~x - ~y) => (tan(~x) - tan(~y)) / (1 + tan(~x) * tan(~y))
@rule csc(~x - ~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(sec(~x) * csc(~y) - csc(~x) * sec(~y))
@rule sec(~x - ~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(csc(~x) * csc(~y) + sec(~x) * sec(~y))
@rule csc(~x -
~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(sec(~x) * csc(~y) - csc(~x) * sec(~y))
@rule sec(~x -
~y) => sec(~x) * sec(~y) * csc(~x) * csc(~y) /
(csc(~x) * csc(~y) + sec(~x) * sec(~y))
@rule cot(~x - ~y) => (cot(~x) * cot(~y) + 1) / (cot(~x) - cot(~y))

# @rule sin(2*~x) => 2*sin(~x)*cos(~x)
Expand Down Expand Up @@ -239,8 +249,9 @@ rational_rules = [@rule Ω(+(~~xs)) => sum(map(Ω, ~~xs))
@rule Ω(^(~x::is_poly, ~k::is_neg_int)) => expand(^(
decompose_rational(~x),
-~k))
@rule Ω(~x / ^(~y::is_poly, ~k::is_pos_int)) => expand(~x *
^(
@rule Ω(~x /
^(~y::is_poly, ~k::is_pos_int)) => expand(~x *
^(
decompose_rational(~y),
~k))
@rule Ω(~x / ~y::is_poly) => expand(~x * decompose_rational(~y))
Expand Down
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