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The package @code{abs_integrate} extends Maxima's integration code to
some integrands that involve the absolute value, max, min, signum, or
unit step functions. For integrands of the form @math{p(x) |q(x)|},
where @math{p} is a polynomial and @math{q} is a polynomial that
@code{factor} is able to factor into a product of linear or constant
terms, the @code{abs_integrate} package determines an antiderivative
that is continuous on the entire real line. Additionally, for an
integrand that involves one or more parameters, the function
@code{conditional_integrate} tries to determine an antiderivative that
is valid for all parameter values.


@b{Examples} To use the @code{abs_integrate} package, you'll first need to load it:
@example
 (%i1) load("abs_integrate.mac")$

 (%i2)  integrate(abs(x),x);
 (%o2) (x^2*signum(x))/2
@end example
To convert (%o2) into an expression involving the absolute value function,
apply @code{signum_to_abs}; thus
@example
 (%i3) signum_to_abs(%);
 (%o3) (x*abs(x))/2
@end example
When the integrand has the form @math{p(x) |x - c1| |x - c2| ... |x - cn|},
where @math{p(x)} is a polynomial and @math{c1, c2, ..., cn} are constants,
the @code{abs_integrate} package returns an antiderivative that is valid on the
entire real line; thus @i{without} making assumptions on @math{a} and @math{b};
for example
@example
 (%i4) factor(convert_to_signum(integrate(abs((x-a)*(x-b)),x,a,b)));
 (%o4) ((b-a)^3*signum(b-a)^2)/6
@end example

Additionally, @code{abs_integrate} is able to find antiderivatives of some
integrands involving @code{max, min, signum}, and @code{unit_step};
examples:
@example
 (%i5) integrate(max(x,x^2),x);
 (%o5) signum(x-1)*((x^3*signum(x))/6-1/6)+signum(x-1)*(1/4-(x^2*signum(x))/4)+x^3/6+x^2/4

 (%i6) integrate(signum(x) - signum(1-x),x);
 (%o6) x*signum(x)+(x-1)*signum(x-1)
@end example
A plot indicates that indeed (%o5) and (%o6) are continuous at zero and at one.

For definite integrals with numerical integration limits (including
both minus and plus infinity), the @code{abs_integrate} package
converts the integrand to signum form and then it tries to subdivide
the integration region so that the integrand simplifies to a
non-signum expression on each subinterval; for example
@example
 (%i1) integrate(1 / (1 + abs(x-5)),x,-5,6);
 (%o1) log(11)+log(2)
@end example

Finally, @code{abs_integrate} is able to determine antiderivatives of @emph{some}
functions of the form @math{F(x, |x - a|)}; examples
@example
 (%i2) integrate(1/(1 + abs(x)),x);
 (%o2) ((signum(x)+1)*log(x+1))/2-(log(1-x)*(1-signum(x)))/2

 (%i3) integrate(cos(x + abs(x)),x);
 (%o3) ((signum(x)+1)*sin(2*x)-2*x*signum(x)+2*x)/4
@end example

Barton Willis (Professor of Mathematics, University of Nebraska at
Kearney) wrote the @code{abs_integrate} package and its English
language user documentation. This documentation also describes the
@code{partition} package for integration. Richard Fateman wrote
@code{partition}. Additional documentation for @code{partition} is
located at http://www.cs.berkeley.edu/~fateman/papers/partition.pdf.


@b{To use} @file{load(abs_integrate)}

@defvr {Option} extra_integration_methods
Default value: @code{['signum_int, 'abs_integrate_use_if]}

The list @code{extra_integration_methods} is a list of functions for
integration. When @code{integrate} is unable to find an
antiderivative, Maxima uses the methods in
@code{extra_integration_methods} to attempt to determine an
antiderivative.

Each function @code{f} in @code{extra_integration_methods} should have
the form @code{f(integrand, variable)}. The function @code{f} may
either return @code{false} to indicate failure, or it may return an
expression involving an integration noun form. The integration methods
are tried from the first to the last member of
@code{extra_integration_methods}; when no method returns an expression
that does not involve an integration noun form, the value of the
integral is the last value that does not fail (or a pure noun form if
all methods fail).

When the function @code{abs_integrate_use_if} is successful, it returns
a conditional expression; for example
@example
 (%i2) integrate(1/(1 + abs(x+1) + abs(x-1)),x);
 (%o2) %if(-(x+1)>0,-log(1-2*x)/2+log(3)-2/3,%if(-(x-1)>0,x/3+log(3)/2-1/3,log(2*x+1)/2))

 (%i3) integrate(exp(-abs(x-1) - abs(x)),x);
 (%o3) %if(-x>0,%e^(2*x-1)/2-2*%e^(-1),%if(-(x-1)>0,%e^(-1)*x-(3*%e^(-1))/2,-%e^(1-2*x)/2))
@end example
For definite integration, these conditional expressions can cause trouble:
@example
 (%i4) integrate(exp(-abs(x-1) - abs(x)),x, minf,inf);
 (%o4) limit(%if(-x>0,(%e^(-1)*(%e^(2*x)-4))/2,%if(-(x-1)>0,(%e^(-1)*(2*x-3))/2,-%e^(1-2*x)/2)),x,inf,minus)-limit(%if(-x>0,(%e^(-1)*(%e^(2*x)-4))/2,%if(-(x-1)>0,(%e^(-1)*(2*x-3))/2,-%e^(1-2*x)/2)),x,-inf,plus)
@end example
For such definite integrals, try disallowing the method @code{abs_integrate_use_if}:
@example
(%i9) integrate(exp(-abs(x-1) - abs(x)),x, minf,inf), extra_integration_methods : ['signum_int];
(%o9) 2*%e^(-1)
@end example

@b{Related options} @i{extra_definite_integration_methods}

@b{To use} @file{load(abs_integrate)}


@end defvr

@defvr {Option} extra_definite_integration_methods
Default value: @code{['abs_defint]}

The list @code{extra_definite_integration_methods} is a list of extra
functions for @i{definite} integration.  When @code{integrate} is
unable to find a definite integral, Maxima uses the methods in
@code{extra_definite_integration_methods} to attempt to determine an
antiderivative.

Each function @code{f} in @code{extra_definite_integration_methods}
should have the form @code{f(integrand, variable, lo, hi)}, where
@code{lo} and @code{hi} are the lower and upper limits of integration,
respectively.  The function @code{f} may either return @code{false} to
indicate failure, or it may return an expression involving an
integration noun form. The integration methods are tried from the
first to the last member of @code{extra_definite_integration_methods};
when no method returns an expression that does not involve an
integration noun form, the value of the integral is the last value
that does not fail (or a pure noun form if all methods fail).


@b{Related options} @i{extra_integration_methods}

@b{To use} @file{load(abs_integrate)}

@end defvr

@deffn {Function} intfudu(@var{e}, @var{x})

This function uses the derivative divides rule for integrands of the
form @math{f(w(x)) * diff(w(x),x)} When @code{infudu} is unable to find
an antiderivative, it returns false.

@example
(%i1) intfudu(cos(x^2) * x,x);
(%o1) sin(x^2)/2

(%i3) intfudu(x * sqrt(1+x^2),x);
(%o3) (x^2+1)^(3/2)/3

(%i4) intfudu(x * sqrt(1 + x^4),x);
(%o4) false
@end example

For the last example, the derivative divides rule fails, so
@code{intfudu} returns false. 

A hashed array @code{intable} contains the antiderivative data. To append a fact
to the hash table, say @math{integrate(f) = g}, do this:

@example
(%i1) intable[f] : lambda([u],  [g(u),diff(u,%voi)]);
(%o1) lambda([u],[g(u),diff(u,%voi)])

(%i2) intfudu(f(z),z);
(%o2) g(z)

(%i3) intfudu(f(w(x)) * diff(w(x),x),x);
(%o3) g(w(x))
@end example

An alternative to calling @code{intfudu} directly is to use the @code{extra_integration_methods}
mechanism; an example:
@example
(%i1) load("abs_integrate.mac")$
(%i2) load(basic)$
(%i3) load("partition.mac")$

(%i4) integrate(bessel_j(1,x^2) * x,x);
(%o4) integrate(bessel_j(1,x^2)*x,x)

(%i5) push('intfudu, extra_integration_methods)$

(%i6) integrate(bessel_j(1,x^2) * x,x);
(%o6) -bessel_j(0,x^2)/2
@end example

@b{To use} @file{load(partition)}

@b{Additional documentation} http://www.cs.berkeley.edu/~fateman/papers/partition.pdf

@b{Related functions} @i{intfugudu}

@end deffn

@deffn {Function} intfugudu (@var{e}, @var{x})

This function uses the derivative divides rule for integrands of the
form @math{f(w(x)) * g(w(x)) * diff(w(x),x)}. When @code{infudu} is
unable to find an antiderivative, it returns false. 

@example
(%i1) diff(jacobi_sn(x,2/3),x);
(%o1) jacobi_cn(x,2/3)*jacobi_dn(x,2/3)

(%i2) intfugudu(%,x);
(%o2) jacobi_sn(x,2/3)

(%i3) diff(jacobi_dn(x^2,a),x);
(%o3) -2*a*x*jacobi_cn(x^2,a)*jacobi_sn(x^2,a)

(%i4) intfugudu(%,x);
(%o4) jacobi_dn(x^2,a)
@end example

For a method for automatically calling @code{infugudu} from @code{integrate}, see
the documentation for @code{intfudu}.

@b{To use} @file{load(partition)}

@b{Additional documentation} http://www.cs.berkeley.edu/~fateman/papers/partition.pdf

@b{Related functions} @i{intfudu}

@end deffn

@deffn {Function} signum_to_abs (@var{e})
 
This function replaces subexpressions of the form @math{q signum(q)} by
@math{abs(q)}.  Before it does these substitutions, it replaces
subexpressions of the form @math{signum(p) * signum(q)} by
@math{signum(p * q)}; examples:

@example
 (%i1) map('signum_to_abs, [x * signum(x), x * y * signum(x)* signum(y)/2]);
 (%o1) [abs(x),(abs(x)*abs(y))/2]
@end example
@b{To use} @file{load(abs_integrate)}

@end deffn

@deffn {Function} conditional_integrate (@var{e}, @var{x})

For an integrand with one or more parameters, this function tries to determine an
antiderivative that is valid for all parameter values. When successful, this
function returns a conditional expression for the antiderivative.

@example
(%i1) conditional_integrate(cos(m*x),x);
(%o1) %if(m#0,sin(m*x)/m,x)

(%i2) conditional_integrate(cos(m*x)*cos(x),x);
(%o2) %if((m-1#0) %and (m+1#0),-((m-1)*cos((m+1)*x)+(m+1)*cos((m-1)*x))/(2*m^2-2),(sin(2*x)+2*x)/4)

(%i3) sublis([m=6],%);
(%o3) -(5*cos(7*x)+7*cos(5*x))/70

(%i4) conditional_integrate(exp(a*x^2+b*x),x);
(%o4) %if(a#0,-(sqrt(%pi)*%e^(-b^2/(4*a))*erf((2*a*x+b)/(2*sqrt(-a))))/(2*sqrt(-a)),%if(b#0,%e^(b*x)/b,x))
@end example

@end deffn

@deffn {Function} convert_to_signum (@var{e})

This function replaces subexpressions of the form @math{abs(q), unit_step(q),
 min(q1,q2, ..., qn)} and @math{max(q1,q2, ..., qn)} by equivalent @math{signum}
terms.

@example
 (%i1) map('convert_to_signum, [abs(x), unit_step(x), max(a,2), min(a,2)]);
 (%o1) [x*signum(x),(signum(x)+1)/2,((a-2)*signum(a-2)+a+2)/2,(-(a-2)*signum(a-2)+a+2)/2]
@end example

To convert @code{unit_step} to signum form, the function @code{convert_to_signum} 
uses @math{unit_step(x) = (1 + signum(x))/2}. 

@b{To use} @file{load(abs_integrate)}

@b{Related functions} @i{signum_to_abs}

@end deffn

