Customizing the Integrator
--------------------------

Calc has two built-in rewrite rules called `IntegRules' and
`IntegAfterRules' which you can edit to define new integration
methods.  See Rewrite Rules.  At each step of the integration
process, Calc wraps the current integrand in a call to the fictitious
function `integtry(EXPR,VAR)', where EXPR is the integrand and VAR is
the integration variable.  If your rules rewrite this to be a plain
formula (not a call to `integtry'), then Calc will use this formula as
the integral of EXPR.  For example, the rule `integtry(mysin(x),x) :=
-mycos(x)' would define a rule to integrate a function `mysin' that
acts like the sine function.  Then, putting `4 mysin(2y+1)' on the
stack and typing `a i y' will produce the integral `-2 mycos(2y+1)'.
Note that Calc has automatically made various transformations on the
integral to allow it to use your rule; integral tables generally give
rules for `mysin(a x + b)', but you don't need to use this much
generality in your `IntegRules'.

As a more serious example, the expression `exp(x)/x' cannot be
integrated in terms of the standard functions, so the "exponential
integral" function `Ei(x)' was invented to describe it.  We can get
Calc to do this integral in terms of a made-up `Ei' function by adding
the rule `[integtry(exp(x)/x, x) := Ei(x)]' to `IntegRules'.  Now
entering `exp(2x)/x' on the stack and typing `a i x' yields `Ei(2 x)'.
This new rule will work with Calc's various built-in integration
methods (such as integration by substitution) to solve a variety of
other problems involving `Ei': For example, now Calc will also be able
to integrate `exp(exp(x))' and `ln(ln(x))' (to get `Ei(exp(x))' and `x
ln(ln(x)) - Ei(ln(x))', respectively).

Your rule may do further integration by calling `integ'.  For example,
`integtry(twice(u),x) := twice(integ(u))' allows Calc to integrate
`twice(sin(x))' to get `twice(-cos(x))'.  Note that `integ' was called
with only one argument.  This notation is allowed only within
`IntegRules'; it means "integrate this with respect to the same
integration variable."  If Calc is unable to integrate `u', the
integration that invoked `IntegRules' also fails.  Thus integrating
`twice(f(x))' fails, returning the unevaluated integral
`integ(twice(f(x)), x)'.  It is still legal to call `integ' with two
or more arguments, however; in this case, if `u' is not integrable,
`twice' itself will still be integrated: If the above rule is changed
to `... := twice(integ(u,x))', then integrating `twice(f(x))' will
yield `twice(integ(f(x),x))'.

If a rule instead produces the formula `integsubst(SEXPR, SVAR)',
either replacing the top-level `integtry' call or nested anywhere
inside the expression, then Calc will apply the substitution `U =
SEXPR(SVAR)' to try to integrate the original EXPR.  For example, the
rule `sqrt(a) := integsubst(sqrt(x),x)' says that if Calc ever finds a
square root in the integrand, it should attempt the substitution `u =
sqrt(x)'.  (This particular rule is unnecessary because Calc always
tries "obvious" substitutions where SEXPR actually appears in the
integrand.)  The variable SVAR may be the same as the VAR that
appeared in the call to `integtry', but it need not be.

When integrating according to an `integsubst', Calc uses the equation
solver to find the inverse of SEXPR (if the integrand refers to VAR
anywhere except in subexpressions that exactly match SEXPR).  It uses
the differentiator to find the derivative of SEXPR and/or its inverse
(it has two methods that use one derivative or the other).  You can
also specify these items by adding extra arguments to the `integsubst'
your rules construct; the general form is `integsubst(SEXPR, SVAR,
SINV, SPRIME)', where SINV is the inverse of SEXPR (still written as a
function of SVAR), and SPRIME is the derivative of SEXPR with respect
to SVAR.  If you don't specify these things, and Calc is not able to
work them out on its own with the information it knows, then your
substitution rule will work only in very specific, simple cases.

Calc applies `IntegRules' as if by `C-u 1 a r IntegRules'; in other
words, Calc stops rewriting as soon as any rule in your rule set
succeeds.  (If it weren't for this, the `integsubst(sqrt(x),x)'
example above would keep on adding layers of `integsubst' calls
forever!)

Another set of rules, stored in `IntegSimpRules', are applied every
time the integrator uses `a s' to simplify an intermediate result.
For example, putting the rule `twice(x) := 2 x' into `IntegSimpRules'
would tell Calc to convert the `twice' function into a form it knows
whenever integration is attempted.

One more way to influence the integrator is to define a function with
the `Z F' command (See Algebraic Definitions).  Calc's integrator
automatically expands such functions according to their defining
formulas, even if you originally asked for the function to be left
unevaluated for symbolic arguments.  (Certain other Calc systems, such
as the differentiator and the equation solver, also do this.)

Sometimes Calc is able to find a solution to your integral, but it
expresses the result in a way that is unnecessarily complicated.  If
this happens, you can either use `integsubst' as described above to
try to hint at a more direct path to the desired result, or you can
use `IntegAfterRules'.  This is an extra rule set that runs after the
main integrator returns its result; basically, Calc does an `a r
IntegAfterRules' on the result before showing it to you.  (It also
does an `a s', without `IntegSimpRules', after that to further
simplify the result.)  For example, Calc's integrator sometimes
produces expressions of the form `ln(1+x) - ln(1-x)'; the default
`IntegAfterRules' rewrite this into the more readable form `2
arctanh(x)'.  Note that, unlike `IntegRules', `IntegSimpRules' and
`IntegAfterRules' are applied any number of times until no further
changes are possible.  Rewriting by `IntegAfterRules' occurs only
after the main integrator has finished, not at every step as for
`IntegRules' and `IntegSimpRules'.