AngouriMath
SolveAPowerOfAFunctionOfAnExponentialBesideXByAnAnsatz(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
A power that is not whole of a rational function of one exponential, W = B(t)^p with
t = e^(s x) , beside a polynomial in x whose coefficients are rational in t, by the ansatz
F = W P/(t^m E) : P a polynomial in x and t of unknown coefficients, and E read
off the denominator of the rest,R = f/W . Rubi's 6.7.1
sqrt(csch(x)) (x cosh(x) - 4 sech(x) tanh(x)) is
(2 x/sqrt(csch(x)) - 4 sech(x)/csch(x)^(3/2))' , sqrt(csch(x)) times a rational
function of x ande^x . By parts leaves -2/sqrt(csch(x)) , which is elliptic, and so is
the other term, and the split declined them both.
off the denominator of the rest,
function of x and
the other term, and the split declined them both.
Remarks
denominator, linear in the unknowns. By Liouville's theorem an elementary antiderivative with no
logarithmic part is W times a rational function of x and t, with a denominator free of x, as R's
is, and a degree in x no higher than R's: a higher coefficient would be a constant over W, which
is not rational. Its denominator is R's with each factor's power lowered by one, or R's itself,
with t to any power, as SolveByRischNormanAnsatz(AngouriMath.Entity,AngouriMath.Entity.Variable) bounds its own.
power of the base, which holds on every branch:
same integrand was right only where
by its derivative.
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