AngouriMath
SolveByPartsAgainstAnInverseHyperbolicFunctionAsItsLogarithm(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)
Method (no overloads)
Summary
By parts against an inverse hyperbolic function written as the logarithm the parser makes of
it, the logarithm's derivative written as the algebraic function it is:
ln(A + sqrt(A^2 + k))' = A'/sqrt(A^2 + k) for a constant k -- asinh ,
acosh , asech and acsch -- and ln((1 + u)/(1 - u))' = 2 u'/(1 - u^2) ,
the same for(u + 1)/(u - 1) -- atanh and acoth . With the integrand
P + Q L for the logarithm L and G = integral Q , it is
integral P + G L - integral G L' . Rubi's 7.1-7.6, x (a + b acosh(c x))/(d + e x^2)^2 and asech(a + b x)/x^3 , which timed out.
https://github.com/asc-community/AngouriMath/issues/233
it, the logarithm's derivative written as the algebraic function it is:
the same for
https://github.com/asc-community/AngouriMath/issues/233
Remarks
Differentiated as written, the logarithm is a quotient with the root in both parts, which the
search never brought down to the root alone. The derivatives are identities of the algebra
with no branch in them. Only for aQ that is rational and not a polynomial: beside a
polynomial the inverse function is answered already, and asked here would come back respelled;
beside a root the rules for the inverse function answer, and asking the root's integral here
first took the time they answer in. Not
re-entered, and not whereG holds L itself, whose question is the one asked:
asinh(x)/sqrt(x^2 + 1) is L L' . Nor where the answer holds on part of the
line only, an inequality in x among its conditions: Rubi's 7.5.1(a + b asech(c x))/x^n is answered for every x without this rule. Nor where a factor below Q 's bar divides the
function's own quadratic,1 - u^2 or the radicand, as in atanh(a x)/(1 - a^2 x^2)^3 :
the substitution under the function answers those, and by parts here ran out of time. And an
artanh or arcoth only of a linear: besideatanh(x sqrt(e)/sqrt(d + e x^2))/x^n the
remainder was slow to decline.
search never brought down to the root alone. The derivatives are identities of the algebra
with no branch in them. Only for a
polynomial the inverse function is answered already, and asked here would come back respelled;
beside a root the rules for the inverse function answer, and asking the root's integral here
first took the time they answer in. Not
re-entered, and not where
line only, an inequality in x among its conditions: Rubi's 7.5.1
function's own quadratic,
the substitution under the function answers those, and by parts here ran out of time. And an
artanh or arcoth only of a linear: beside
remainder was slow to decline.
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