AngouriMath
SolveByExponentialAnsatz(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
An integrand e^h R , with R rational in x and h rational
inx or absent, integrated by the ansatz F = e^h N/D : D read
off the denominator ofR , N a polynomial of unknown coefficients, and
F' = e^h R a linear system in them.
in
off the denominator of
Remarks
None had an antiderivative: the first two are not a polynomial times an exponential,
which is the shape by parts reads, and the third has a denominator nothing factors.
Each is the derivative of something of the same shape, and that is what is looked
for -- Liouville's theorem says the elementary antiderivative of
there is one, is
fails because there is no such antiderivative and not because the shape was wrong.
For
logarithmic part required to vanish; a denominator with a repeated factor and a
logarithmic part beside it is left to the splits, as before.
denominator by one, so
written power lowered by one, and then the denominator itself, and
denominator is
((N' D - N D') q^2 + (p' q - p q') N D) D_R = N_R D^2 q^2
the same elimination the symbolic partial-fraction split uses and checked the same way
before anything is returned: at sampled points with every symbol pinned. The degree
of
an unknown the system does not need is zero.
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