AngouriMath
SolveAHyperbolicIntegrandAsATrigonometricOne(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean,System.Boolean,System.Boolean,System.Boolean)
Method (no overloads)
Summary
A hyperbolic integrand asked as a trigonometric one, by Osborn's rule: with L = i s ,
cosh(L) = cos(s) , sinh(L) = i sin(s) , tanh(L) = i tan(s) and
coth(L) = -i cot(s) , and dx = i ds/b for L = a + b x . The parser writes
the hyperbolic functions as exponentials --cosh(L) is (e^L + e^(-L))/2 -- so
the trigonometric rules never read them:(b cosh(x) + c sinh(x) + sqrt(b^2 - c^2))^(1/2) timed out where (b cos(x) + c sin(x) + sqrt(b^2 + c^2))^(1/2) is answered at once, and
as(b cos(s) + i c sin(s) + sqrt(b^2 - c^2))^(1/2) it is the same question. The answer
ins is written back with s = -i L and checked against the integrand at
sampled points: a trigonometric answer derived for a reals can take a different
branch on the imaginary axis, and then the rule declines. Rubi's 6.7.1.
https://github.com/asc-community/AngouriMath/issues/233
the hyperbolic functions as exponentials --
the trigonometric rules never read them:
as
in
sampled points: a trigonometric answer derived for a real
branch on the imaginary axis, and then the rule declines. Rubi's 6.7.1.
https://github.com/asc-community/AngouriMath/issues/233
Remarks
Only where a sine or cosine is among the quotients, every exponential of x is a whole power
of onee^L inside one of them, and x stands nowhere else. With
rationalToo false, only where the integrand is not a rational function of
e^L , which the exponential substitution answers in real terms; with it true, only those,
after that substitution has declined them. Asked only of the question asked, and last: the
question ins is not cheap to decline, and asked early it took time from the rules
that answerx/cosh(x)^(7/2) + 3/5 x sqrt(cosh(x)) and the like. Nor in the first pass
where the exponential substitution wants a sum over roots and the question ins , a
rational function ofcos(s) or sin(s) alone, wants one too, by a factor below
the bar of degree above two: the second pass answers it ine^L , and the question in
s took seconds to decline, five gigabytes for 1/(1 + sinh(x)^3) against 0.2.
of one
after that substitution has declined them. Asked only of the question asked, and last: the
question in
that answer
where the exponential substitution wants a sum over roots and the question in
rational function of
the bar of degree above two: the second pass answers it in
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