AngouriMath
SolveByInverseHyperbolicSubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
An integrand holding an inverse hyperbolic function of a linear in the variable,
integrated by the substitution that undoes it. The inverse hyperbolic functions
are not nodes here --arsinh(L) is written ln(L + sqrt(L^2 + 1)) ,
arcosh(L) is ln(L + sqrt(L^2 - 1)) and artanh(L) is
ln((1 + L)/(1 - L))/2 -- so the logarithm is read for the function it is,
and underL = sinh(u) , cosh(u) or tanh(u) it is u , the
radical ofL^2 + 1 is cosh(u) , of L^2 - 1 is sinh(u) (for u not negative, where the principal arcosh lies) and
1 - L^2 is sech(u)^2 ; a constant multiple of the quadratic is the
multiple's power times that, the generic case. What is left is a function of
u and of exponentials of it, which the exponential rules read: Rubi's
x arsinh(a x) is u sinh(2u)/(2 a^2) , parts once, where parts in
x left the logarithm's derivative as a quotient of radicals.
The way back writese^u as L + sqrt(L^2 + 1) and e^(-u) as
sqrt(L^2 + 1) - L , on the principal branch.
arsech(L) and arcsch(L) are the arcosh and arsinh of
1/L , and so u under L = sech(u) and L = csch(u) , with
e^u as 1/L + sqrt(1/L^2 - 1) and 1/L + sqrt(1/L^2 + 1) : Rubi's
1/(x^2 (a + b arcsch(c x))) is -c cosh(u)/(a + b u) , which the
hyperbolic sine and cosine integrals answer.
https://github.com/asc-community/AngouriMath/issues/718
integrated by the substitution that undoes it. The inverse hyperbolic functions
are not nodes here --
and under
radical of
multiple's power times that, the generic case. What is left is a function of
The way back writes
hyperbolic sine and cosine integrals answer.
https://github.com/asc-community/AngouriMath/issues/718
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