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OffItsRestrictions(AngouriMath.Entity,AngouriMath.Entity,System.Collections.Generic.List{AngouriMath.Entity},System.Boolean,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

answer, an antiderivative of integrand where
restrictions hold, as one wherever the integrand has a value, with the
condition it still holds under, null where it needs none: as it is
where it holds on every interval between the roots of the restrictions, with the sign
of the one restriction in front where that is what it is off it, and under the
restrictions otherwise. byParity where answer is
the antiderivative for positive x written in |x|.

Remarks

sqrt(1 - u^2)/sqrt(u - 1) is sqrt(-(1 + u)) for u > 1 and its
negative for -1 < u < 1, where the root below the bar is imaginary, so an
antiderivative found through the one is the negative of one off it, and
sgn(u - 1) F holds on both sides. Rubi writes that sign as
sqrt(1 - c^2 x^2)/(c x sqrt(1 - 1/(c^2 x^2))), which is i sgn(c x) sgn(c^2 x^2 - 1).
Only where the integrand has no value at the restriction's roots, which then part its
domain: where it has one, an antiderivative is continuous there and sgn(P) F is not.
Decided at two points of every interval between the real roots of the restrictions and
of x, comparing complex values, every other symbol pinned to an exact rational so that
the roots are exact. An interval where the two cannot be compared does not hold. The
derivative is compared with each (P^(1/n))^n in the answer written P, which
is the same number on every branch: (sqrt(x^2 - 1))^2 under a root, inside
(-1, 1), is an interval about the negative axis that the evaluation cannot place
on either side of the cut.
https://github.com/asc-community/AngouriMath/issues/233

AngouriMath

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AngouriMath

OffItsRestrictions​(AngouriMath.​Entity,​AngouriMath.​Entity,​System.​Collections.​Generic.​List{AngouriMath.​Entity},​System.​Boolean,​AngouriMath.​Entity.​Variable)

OffItsRestrictions(AngouriMath.Entity,AngouriMath.Entity,System.Collections.Generic.List{AngouriMath.Entity},System.Boolean,AngouriMath.Entity.Variable)