<s>
In	O
calculus	O
,	O
symbolic	B-Algorithm
integration	I-Algorithm
is	O
the	O
problem	O
of	O
finding	O
a	O
formula	O
for	O
the	O
antiderivative	B-Algorithm
,	O
or	O
indefinite	B-Algorithm
integral	I-Algorithm
,	O
of	O
a	O
given	O
function	O
f(x )	O
,	O
i.e.	O
</s>
<s>
The	O
term	O
symbolic	O
is	O
used	O
to	O
distinguish	O
this	O
problem	O
from	O
that	O
of	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
where	O
the	O
value	O
of	O
F	O
is	O
sought	O
at	O
a	O
particular	O
input	O
or	O
set	O
of	O
inputs	O
,	O
rather	O
than	O
a	O
general	O
formula	O
for	O
F	O
.	O
</s>
<s>
Both	O
problems	O
were	O
held	O
to	O
be	O
of	O
practical	O
and	O
theoretical	O
importance	O
long	O
before	O
the	O
time	O
of	O
digital	O
computers	O
,	O
but	O
they	O
are	O
now	O
generally	O
considered	O
the	O
domain	O
of	O
computer	B-General_Concept
science	I-General_Concept
,	O
as	O
computers	O
are	O
most	O
often	O
used	O
currently	O
to	O
tackle	O
individual	O
instances	O
.	O
</s>
<s>
Many	O
expressions	O
which	O
are	O
relatively	O
simple	O
do	O
not	O
have	O
integrals	B-Algorithm
that	O
can	O
be	O
expressed	O
in	O
closed	O
form	O
.	O
</s>
<s>
See	O
antiderivative	B-Algorithm
and	O
nonelementary	O
integral	O
for	O
more	O
details	O
.	O
</s>
<s>
A	O
procedure	O
called	O
the	O
Risch	O
algorithm	O
exists	O
which	O
is	O
capable	O
of	O
determining	O
whether	O
the	O
integral	O
of	O
an	O
elementary	O
function	O
(	O
function	O
built	O
from	O
a	O
finite	O
number	O
of	O
exponentials	O
,	O
logarithms	O
,	O
constants	O
,	O
and	O
nth	O
roots	O
through	O
composition	B-Application
and	O
combinations	O
using	O
the	O
four	O
elementary	O
operations	O
)	O
is	O
elementary	O
and	O
returning	O
it	O
if	O
it	O
is	O
.	O
</s>
<s>
It	O
was	O
first	O
implemented	O
in	O
Reduce	B-Language
in	O
the	O
case	O
of	O
purely	O
transcendental	O
functions	O
;	O
the	O
case	O
of	O
purely	O
algebraic	O
functions	O
was	O
solved	O
and	O
implemented	O
in	O
Reduce	B-Language
by	O
James	O
H	O
.	O
Davenport	O
;	O
the	O
general	O
case	O
was	O
solved	O
by	O
Manuel	O
Bronstein	O
,	O
who	O
implemented	O
almost	O
all	O
of	O
it	O
in	O
Axiom	B-Language
,	O
though	O
to	O
date	O
there	O
is	O
no	O
implementation	O
of	O
the	O
Risch	O
algorithm	O
which	O
can	O
deal	O
with	O
all	O
of	O
the	O
special	O
cases	O
and	O
branches	O
in	O
it	O
.	O
</s>
<s>
However	O
,	O
the	O
Risch	O
algorithm	O
applies	O
only	O
to	O
indefinite	B-Algorithm
integrals	I-Algorithm
,	O
while	O
most	O
of	O
the	O
integrals	B-Algorithm
of	O
interest	O
to	O
physicists	O
,	O
theoretical	O
chemists	O
,	O
and	O
engineers	O
are	O
definite	B-Algorithm
integrals	I-Algorithm
often	O
related	O
to	O
Laplace	O
transforms	O
,	O
Fourier	B-Algorithm
transforms	I-Algorithm
,	O
and	O
Mellin	O
transforms	O
.	O
</s>
<s>
Lacking	O
a	O
general	O
algorithm	O
,	O
the	O
developers	O
of	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
have	O
implemented	O
heuristics	B-Algorithm
based	O
on	O
pattern-matching	B-Language
and	O
the	O
exploitation	O
of	O
special	O
functions	O
,	O
in	O
particular	O
the	O
incomplete	O
gamma	O
function	O
.	O
</s>
<s>
Although	O
this	O
approach	O
is	O
heuristic	B-Algorithm
rather	O
than	O
algorithmic	O
,	O
it	O
is	O
nonetheless	O
an	O
effective	O
method	O
for	O
solving	O
many	O
definite	B-Algorithm
integrals	I-Algorithm
encountered	O
by	O
practical	O
engineering	O
applications	O
.	O
</s>
<s>
Earlier	O
systems	O
such	O
as	O
Macsyma	B-Language
had	O
a	O
few	O
definite	B-Algorithm
integrals	I-Algorithm
related	O
to	O
special	O
functions	O
within	O
a	O
look-up	O
table	O
.	O
</s>
<s>
However	O
this	O
particular	O
method	O
,	O
involving	O
differentiation	O
of	O
special	O
functions	O
with	O
respect	O
to	O
its	O
parameters	O
,	O
variable	O
transformation	O
,	O
pattern	B-Language
matching	I-Language
and	O
other	O
manipulations	O
,	O
was	O
pioneered	O
by	O
developers	O
of	O
the	O
Maple	B-Language
system	O
and	O
then	O
later	O
emulated	O
by	O
Mathematica	B-Language
,	O
Axiom	B-Language
,	O
MuPAD	B-Language
and	O
other	O
systems	O
.	O
</s>
<s>
The	O
main	O
problem	O
in	O
the	O
classical	O
approach	O
of	O
symbolic	B-Algorithm
integration	I-Algorithm
is	O
that	O
,	O
if	O
a	O
function	O
is	O
represented	O
in	O
closed	O
form	O
,	O
then	O
,	O
in	O
general	O
,	O
its	O
antiderivative	B-Algorithm
has	O
not	O
a	O
similar	O
representation	O
.	O
</s>
<s>
This	O
includes	O
the	O
computation	O
of	O
antiderivatives	B-Algorithm
and	O
definite	B-Algorithm
integrals	I-Algorithm
(	O
this	O
amounts	O
to	O
evaluating	O
the	O
antiderivative	B-Algorithm
at	O
the	O
endpoints	O
of	O
the	O
interval	O
of	O
integration	O
)	O
.	O
</s>
<s>
This	O
includes	O
also	O
the	O
computation	O
of	O
the	O
asymptotic	O
behavior	O
of	O
the	O
function	O
at	O
infinity	O
,	O
and	O
thus	O
the	O
definite	B-Algorithm
integrals	I-Algorithm
on	O
unbounded	O
intervals	O
.	O
</s>
<s>
All	O
these	O
operations	O
are	O
implemented	O
in	O
the	O
algolib	O
library	O
for	O
Maple	B-Language
.	O
</s>
<s>
is	O
a	O
symbolic	O
result	O
for	O
an	O
indefinite	B-Algorithm
integral	I-Algorithm
(	O
here	O
C	O
is	O
a	O
constant	O
of	O
integration	O
)	O
,	O
</s>
<s>
is	O
a	O
numerical	O
result	O
for	O
the	O
same	O
definite	B-Algorithm
integral	I-Algorithm
.	O
</s>
