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Calc is capable of performing some simplifications which may sometimes
be desired but which are not “safe” in all cases. The `a e`
(`calc-simplify-extended`

) [`esimplify`

] command
applies the algebraic simplifications as well as these extended, or
“unsafe”, simplifications. Use this only if you know the values in
your formula lie in the restricted ranges for which these
simplifications are valid. You can use Extended Simplification mode
(`m E`) to have these simplifications done automatically.

The symbolic integrator uses these extended simplifications; one effect
of this is that the integrator’s results must be used with caution.
Where an integral table will often attach conditions like “for positive
‘`a`’ only,” Calc (like most other symbolic integration programs)
will simply produce an unqualified result.

Because `a e`’s simplifications are unsafe, it is sometimes better
to type `C-u -3 a v`, which does extended simplification only
on the top level of the formula without affecting the sub-formulas.
In fact, `C-u -3 j v` allows you to target extended simplification
to any specific part of a formula.

The variable `ExtSimpRules`

contains rewrites to be applied when
the extended simplifications are used. These are applied in addition to
`EvalRules`

and `AlgSimpRules`

. (The `a r AlgSimpRules`
step described above is simply followed by an `a r ExtSimpRules` step.)

Following is a complete list of the “unsafe” simplifications.

Inverse trigonometric or hyperbolic functions, called with their
corresponding non-inverse functions as arguments, are simplified.
For example, ‘` arcsin(sin(x))`’ changes
to ‘

Powers of powers ‘`(x^a)^b`’ are simplified to
‘`x^(a b)`’
for all ‘`a`’ and ‘`b`’. These results will be valid only
in a restricted range of ‘`x`’; for example, in
‘`(x^2)^1:2`’
the powers cancel to get ‘`x`’, which is valid for positive values
of ‘`x`’ but not for negative or complex values.

Similarly, ‘` sqrt(x^a)`’ and ‘

Forms like ‘` sqrt(1 - sin(x)^2)`’ are simplified to, e.g.,
‘

`sin`

,
`cos`

, `tan`

, `sinh`

, and `cosh`

.
Arguments of square roots are partially factored to look for
squared terms that can be extracted. For example,
‘` sqrt(a^2 b^3 + a^3 b^2)`’ simplifies to
‘

The simplifications of ‘` ln(exp(x))`’,
‘

Common factors are canceled from products on both sides of an
equation, even if those factors may be zero: ‘`a x / b x`’
to ‘`a / b`’. Such factors are never canceled from
inequalities: Even the extended simplifications are not bold enough to
reduce ‘`a x < b x`’ to ‘`a < b`’ (or ‘`a > b`’, depending
on whether you believe ‘`x`’ is positive or negative).
The `a M /` command can be used to divide a factor out of
both sides of an inequality.