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SolveByProductToSum​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

 Method (no overloads)

Summary

A product of sines and cosines of different arguments, rewritten as a sum by the
product-to-sum identities and then integrated term by term.

Remarks

sin(A)sin(B) = (cos(A - B) - cos(A + B))/2,
cos(A)cos(B) = (cos(A - B) + cos(A + B))/2,
sin(A)cos(B) = (sin(A + B) + sin(A - B))/2. Each application turns two factors
into a sum of two single ones, and a sine or cosine of something linear in the variable
is a rule the integrator already has — so the whole family comes out at once.
sin(x)sin(2x), cos(x)cos(2x), cos(3x)sin(2x) and
sin(x)sin(2x)sin(3x) had no antiderivative between them.
Why this rather than a substitution. These are the shapes every substitution in
the chain declines, and rightly: there is no inner function to substitute for. A
product of trigonometric functions of unequal arguments is not a function of any one of
them, which is what SolveByHalfAngleSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) notices when it finds an
x left over after rewriting — sin(2x) is not sin(x). The identity
is the tool, not a change of variable.
It terminates. Each rewrite replaces two trigonometric factors with a sum whose
terms hold one each, so the number of such factors in any one product strictly
decreases, and the recursion is over strictly simpler products. Arguments equal to each
other are left alone, since sin(A)^2 is a power rather than a product of two
arguments and wants a reduction formula instead.
Nothing is assumed and no condition is owed. These are identities on the whole
complex plane, not rewrites that hold on an interval: both sides are entire, so unlike
every substitution here the answer is an antiderivative everywhere the integrand is,
with no branch to pick and no interval to be inside.
 https://github.com/asc-community/AngouriMath/issues/718

























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