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

 Method (no overloads)

Summary

An integrand that is a function of e^x with a root in it, integrated by the
substitution u = tanh(x), the hyperbolic counterpart of the tangent's.

Remarks

The library writes the hyperbolic functions as exponentials, and a root of one of
them -- Timofeev's cosh(x)(tanh(x) - cosh(2x))/((sinh(x)^2 + sinh(2x)) sqrt(sinh(2x))) -- is under u = e^x a root of (u^4 - 1)/u^2, a quartic that nothing
rationalises, where its trigonometric twin is a rational function of the tangent
with sqrt(2u/(1 - u^2)) in it and is answered. Under u = tanh(x) the
hyperbolic functions are the same shapes: cosh(x) is (1 - u^2)^(-1/2) and sinh(x) is u (1 - u^2)^(-1/2), exactly and for every real x,
since the hyperbolic cosine is positive -- no sign to carry, where the tangent
substitution carries sgn(cos(x)). So e^x is (1 + u)/w and
e^-x is (1 - u)/w for w = sqrt(1 - u^2), the integrand is
rational in u and w apart from its own roots, and dx = du/(1 - u^2).
Every quotient -- each radicand, and the whole -- is brought over one bar with
w^2 = 1 - u^2 reduced and the denominator cleared of w by its
conjugate, and the polynomials cancelled by their greatest common divisor, so that
e^x + e^-x comes out as 2/w and sinh(2x) as 2u/(1 - u^2);
a radicand over a power of 1 - u^2 gives that power up to w, where it
meets the rest, and a radicand that is a power of 1 - u^2 and w alone
is a root of 1 - u^2 of another index -- sech(x)^(3/4) is
(1 - u^2)^(3/8).
 With a root in the integrand whose radicand holds exponentials of both signs
-- as a Laurent polynomial in e^x, a polynomial over a power with a power on
either side of it, or a rational function over anything else -- which is where
u = e^x leaves a root of a quartic or of a quotient; with one sign only,
sqrt(e^x - 1) or sqrt(1 + tanh(x)), the exponential substitution answers
with a root of a linear, and is left to. And, with a symbol among the coefficients,
for an integrand that is a rational function of tanh(x) -- one in which
w cancels out -- with its roots, if any, of rational functions of it too:
1/(a + b coth(x)^2)^2 is u^4/((b + a u^2)^2 (1 - u^2)) here, where under
u = e^x it is a rational function of a symbolic palindromic quartic
(a + b) u^4 + 2(b - a) u^2 + (a + b) squared, whose roots are what the budget
went on; Rubi's route for the hyperbolic tangent, cotangent, secant and cosecant
families. With rational coefficients the exponential substitution's quartics are
factored over the rationals and answered, and are left to it.
 https://github.com/asc-community/AngouriMath/issues/718

























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