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FriCASIntegration

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Editor: test1 Time: 2014/06/03 17:53:47 GMT+0
Note:

changed:
-
FriCAS inherited from Axiom implementation of Risch algorithm for elementary integration.  It can
handle both easy cases
\begin{axiom}
integrate(x*exp(x^2), x)
integrate(exp(1/x^2)/x^3, x)
\end{axiom}
and some which are hard for simpler algorithms.
For example Maxma 21.1 can not handle
\begin{axiom}
integrate(sqrt(x^2+1)/(x^3+1), x)
\end{axiom}
\begin{axiom}
)set output tex off
)set output algebra on
integrate(sqrt(x^2+1)/(x^4+1), x)
)set output algebra off
)set output tex on
\end{axiom}

None of default integrators in Mathematica 8.0 , Maple 15 and Maxima 21.1 can do
\begin{axiom}
integrate(((exp(x)-x^2+2*x)/(x^2*(exp(x)+x)^2))*exp((x^2-1)/x+1/(exp(x)+x)), x)
integrate(x^6/sqrt((x^7+1)*(x^7+2)), x)
\end{axiom}

FriCAS fixed several bugs in Axiom implementation, for example
\begin{axiom}
integrate(cos(x)^n*sin(x)^(-n-2), x)
integrate(exp(asec(x))/x^2, x)
\end{axiom}
work now but used to return unevaluated result.

Risch algorithm can handle large class of integrands involving special functions
\begin{axiom}
f := D(1/(1 + ellipticE(x^2+a, m)), x)
integrate(f, x)

f := D(1/(Ei(x)-Ei(x-1)), x)
integrate(f, x)

f := D(1/(Si(x+1)+Ci(x-1)), x)
integrate(f, x)

f := D(1/(Si(Ci(x)^2+1)+Ci(Ci(x)^2-1)), x)
integrate(f, x)

f := D(log(Si(Ci(x)^2+1)+Ci(Ci(x)^2-1)), x)
integrate(f, x)
\end{axiom}

Alas, the following does not work
\begin{axiom}
f := D(1/(1 + besselJ(m, x^2+a)), x)
integrate(f, x)
\end{axiom}
Risch algorithm wants to find elementary integral.  For many functions integrals can be written only in
terms of special function.  FriCAS now contains extensions which
can handle some of them, see FriCASSpecialIntegration.

For more elementaty examples see SandboxFriCASTimofeev1, SandboxFriCASTimofeev37,
SandboxFriCASTimofeev5, SandboxFriCASTimofeev9.

FriCAS? inherited from Axiom implementation of Risch algorithm for elementary integration. It can handle both easy cases

fricas

integrate(x*exp(x^2), x)

$\label{eq1}{{e}^{{x}^{2}}}\over 2$

(1)

Type: Union(Expression(Integer),...)

fricas

integrate(exp(1/x^2)/x^3, x)

$\label{eq2}-{{{e}^{1 \over{{x}^{2}}}}\over 2}$

(2)

Type: Union(Expression(Integer),...)

and some which are hard for simpler algorithms. For example Maxma 21.1 can not handle

fricas

integrate(sqrt(x^2+1)/(x^3+1), x)

$\label{eq3}{\left( \begin{array}{@{}l} \displaystyle {{\sqrt{2}}\ {\log{\left({{{{\left({\sqrt{2}}+ x + 1 \right)}\ {\sqrt{{{x}^{2}}+ 1}}}+{{\left(- x - 1 \right)}\ {\sqrt{2}}}-{{x}^{2}}- x - 2}\over{{{\left(x + 1 \right)}\ {\sqrt{{{x}^{2}}+ 1}}}-{{x}^{2}}- x}}\right)}}}+ \ \ \displaystyle {2 \ {\arctan{\left({{{{\left({2 \ {{x}^{2}}}-{2 \ x}+ 1 \right)}\ {\sqrt{{{x}^{2}}+ 1}}}-{2 \ {{x}^{3}}}+{2 \ {{x}^{2}}}-{2 \ x}}\over{x - 1}}\right)}}}- \ \ \displaystyle {2 \ {\arctan \left({{2 \ x}- 1}\right)}}$

(3)

Type: Union(Expression(Integer),...)

fricas

)set output tex off

fricas

)set output algebra on

integrate(sqrt(x^2+1)/(x^4+1), x)

   (4)
       -
             4+-+
            4\|2
         *
            atan
                            +---------+
                            |  +-+
                   +-+      |2\|2  - 4    4+-+
                 (\|2  - 1) |---------  - \|2
                            |  +-+
                           \|2\|2  - 3
              /
                                +---------+
                                |  +-+
                       +-+      |2\|2  - 4
                     (\|2  - 1) |---------
                                |  +-+
                               \|2\|2  - 3
                  *
                     ROOT
                                                      +---------+
                                                      |  +-+
                                       +-+       4+-+ |2\|2  - 4       3 +-+
                                (- 41x\|2  + 58x)\|2  |---------  - 48x \|2
                                                      |  +-+
                                                     \|2\|2  - 3
                              + 
                                   3
                                68x
                           *
                               +------+
                               | 2
                              \|x  + 1
                          + 
                                                              +---------+
                                                              |  +-+
                                 2       +-+      2      4+-+ |2\|2  - 4
                            ((41x  + 17)\|2  - 58x  - 24)\|2  |---------
                                                              |  +-+
                                                             \|2\|2  - 3
                          + 
                                +-+      4+-+2       4      2       +-+      4
                            (24\|2  - 34)\|2   + (48x  + 24x  + 24)\|2  - 68x
                          + 
                                 2
                            - 34x  - 34
                       /
                             +-+
                          24\|2  - 34
                 + 
                                 +---------+
                                 |  +-+      +------+
                        +-+      |2\|2  - 4  | 2
                   (- x\|2  + x) |--------- \|x  + 1
                                 |  +-+
                                \|2\|2  - 3
                 + 
                                 +---------+
                                 |  +-+
                     2 +-+    2  |2\|2  - 4      +-+     4+-+
                   (x \|2  - x ) |---------  + (\|2  - 1)\|2
                                 |  +-+
                                \|2\|2  - 3
     + 
          4+-+
         4\|2
      *
         atan
                         +---------+
                         |  +-+
                +-+      |2\|2  - 4    4+-+
              (\|2  - 1) |---------  + \|2
                         |  +-+
                        \|2\|2  - 3
           /
                             +---------+
                             |  +-+
                    +-+      |2\|2  - 4
                  (\|2  - 1) |---------
                             |  +-+
                            \|2\|2  - 3
               *
                  ROOT
                                                +---------+
                                                |  +-+
                                 +-+       4+-+ |2\|2  - 4       3 +-+      3
                           ((41x\|2  - 58x)\|2  |---------  - 48x \|2  + 68x )
                                                |  +-+
                                               \|2\|2  - 3
                        *
                            +------+
                            | 2
                           \|x  + 1
                       + 
                                                             +---------+
                                                             |  +-+
                                2       +-+      2      4+-+ |2\|2  - 4
                         ((- 41x  - 17)\|2  + 58x  + 24)\|2  |---------
                                                             |  +-+
                                                            \|2\|2  - 3
                       + 
                             +-+      4+-+2       4      2       +-+      4
                         (24\|2  - 34)\|2   + (48x  + 24x  + 24)\|2  - 68x
                       + 
                              2
                         - 34x  - 34
                    /
                          +-+
                       24\|2  - 34
              + 
                              +---------+                          +---------+
                              |  +-+      +------+                 |  +-+
                     +-+      |2\|2  - 4  | 2          2 +-+    2  |2\|2  - 4
                (- x\|2  + x) |--------- \|x  + 1  + (x \|2  - x ) |---------
                              |  +-+                               |  +-+
                             \|2\|2  - 3                          \|2\|2  - 3
              + 
                    +-+     4+-+
                (- \|2  + 1)\|2
     + 
           +-+     4+-+
         (\|2  - 1)\|2
      *
         log
                                     +---------+
                                     |  +-+                         +------+
                      +-+       4+-+ |2\|2  - 4       3 +-+      3  | 2
                ((41x\|2  - 58x)\|2  |---------  - 48x \|2  + 68x )\|x  + 1
                                     |  +-+
                                    \|2\|2  - 3
              + 
                                                    +---------+
                                                    |  +-+
                       2       +-+      2      4+-+ |2\|2  - 4
                ((- 41x  - 17)\|2  + 58x  + 24)\|2  |---------
                                                    |  +-+
                                                   \|2\|2  - 3
              + 
                    +-+      4+-+2       4      2       +-+      4      2
                (24\|2  - 34)\|2   + (48x  + 24x  + 24)\|2  - 68x  - 34x  - 34
           /
                 +-+
              24\|2  - 34
     + 
             +-+     4+-+
         (- \|2  + 1)\|2
      *
         log
                                       +---------+
                                       |  +-+                         +------+
                        +-+       4+-+ |2\|2  - 4       3 +-+      3  | 2
                ((- 41x\|2  + 58x)\|2  |---------  - 48x \|2  + 68x )\|x  + 1
                                       |  +-+
                                      \|2\|2  - 3
              + 
                                                  +---------+
                                                  |  +-+
                     2       +-+      2      4+-+ |2\|2  - 4
                ((41x  + 17)\|2  - 58x  - 24)\|2  |---------
                                                  |  +-+
                                                 \|2\|2  - 3
              + 
                    +-+      4+-+2       4      2       +-+      4      2
                (24\|2  - 34)\|2   + (48x  + 24x  + 24)\|2  - 68x  - 34x  - 34
           /
                 +-+
              24\|2  - 34
  /
                 +---------+
                 |  +-+
        +-+      |2\|2  - 4
     (4\|2  - 4) |---------
                 |  +-+
                \|2\|2  - 3

Type: Union(Expression(Integer),...)

fricas

)set output algebra off

fricas

)set output tex on

None of default integrators in Mathematica 8.0 , Maple 15 and Maxima 21.1 can do

fricas

integrate(((exp(x)-x^2+2*x)/(x^2*(exp(x)+x)^2))*exp((x^2-1)/x+1/(exp(x)+x)), x)

$\label{eq4}{{e}^{{{{\left({{x}^{2}}- 1 \right)}\ {{e}^{x}}}+{{x}^{3}}}\over{{x \ {{e}^{x}}}+{{x}^{2}}}}}\over{{e}^{x}}$

(4)

Type: Union(Expression(Integer),...)

fricas

integrate(x^6/sqrt((x^7+1)*(x^7+2)), x)

$\label{eq5}-{{\log \left({{2 \ {\sqrt{{{x}^{14}}+{3 \ {{x}^{7}}}+ 2}}}-{2 \ {{x}^{7}}}- 3}\right)}\over 7}$

(5)

Type: Union(Expression(Integer),...)

FriCAS? fixed several bugs in Axiom implementation, for example

fricas

integrate(cos(x)^n*sin(x)^(-n-2), x)

$\label{eq6}-{{{\cos \left({x}\right)}\ {\sin \left({x}\right)}\ {{e}^{{\left(- n - 2 \right)}\ {\log \left({\sin \left({x}\right)}\right)}}}\ {{e}^{n \ {\log \left({\cos \left({x}\right)}\right)}}}}\over{n + 1}}$

(6)

Type: Union(Expression(Integer),...)

fricas

integrate(exp(asec(x))/x^2, x)

$\label{eq7}{{\left({{\left(- x - 1 \right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{x}^{2}}+ x - 1 \right)}\ {{e}^{asec \left({x}\right)}}}\over{{2 \ x \ {\sqrt{{{x}^{2}}- 1}}}-{2 \ {{x}^{2}}}}$

(7)

Type: Union(Expression(Integer),...)

work now but used to return unevaluated result.

Risch algorithm can handle large class of integrands involving special functions

fricas

f := D(1/(1 + ellipticE(x^2+a, m)), x)

$\label{eq8}\begin{array}{@{}l} \displaystyle -{{2 \ x \ {\sqrt{-{m \ {{x}^{4}}}-{2 \ a \ m \ {{x}^{2}}}-{{{a}^{2}}\ m}+ 1}}}\over{\left({\left({ \begin{array}{@{}l} \displaystyle {{ellipticE \left({{{{x}^{2}}+ a}, \: m}\right)}^{2}}+ \ \ \displaystyle {2 \ {ellipticE \left({{{{x}^{2}}+ a}, \: m}\right)}}+ \ \ \displaystyle 1$

(8)

Type: Expression(Integer)

fricas

integrate(f, x)

$\label{eq9}1 \over{{ellipticE \left({{{{x}^{2}}+ a}, \: m}\right)}+ 1}$

(9)

Type: Union(Expression(Integer),...)

fricas

f := D(1/(Ei(x)-Ei(x-1)), x)

$\label{eq10}{{{\left(- x + 1 \right)}\ {{e}^{x}}}+{x \ {{e}^{x - 1}}}}\over{\left( \begin{array}{@{}l} \displaystyle {{\left({{x}^{2}}- x \right)}\ {{Ei \left({x}\right)}^{2}}}+{{\left(-{2 \ {{x}^{2}}}+{2 \ x}\right)}\ {Ei \left({x - 1}\right)}\ {Ei \left({x}\right)}}+ \ \ \displaystyle {{\left({{x}^{2}}- x \right)}\ {{Ei \left({x - 1}\right)}^{2}}}$

(10)

Type: Expression(Integer)

fricas

integrate(f, x)

$\label{eq11}1 \over{{Ei \left({x}\right)}-{Ei \left({x - 1}\right)}}$

(11)

Type: Union(Expression(Integer),...)

fricas

f := D(1/(Si(x+1)+Ci(x-1)), x)

$\label{eq12}{{{\left(- x + 1 \right)}\ {\sin \left({x + 1}\right)}}+{{\left(- x - 1 \right)}\ {\cos \left({x - 1}\right)}}}\over{\left( \begin{array}{@{}l} \displaystyle {{\left({{x}^{2}}- 1 \right)}\ {{Si \left({x + 1}\right)}^{2}}}+{{\left({2 \ {{x}^{2}}}- 2 \right)}\ {Ci \left({x - 1}\right)}\ {Si \left({x + 1}\right)}}+ \ \ \displaystyle {{\left({{x}^{2}}- 1 \right)}\ {{Ci \left({x - 1}\right)}^{2}}}$

(12)

Type: Expression(Integer)

fricas

integrate(f, x)

$\label{eq13}1 \over{{Si \left({x + 1}\right)}+{Ci \left({x - 1}\right)}}$

(13)

Type: Union(Expression(Integer),...)

fricas

f := D(1/(Si(Ci(x)^2+1)+Ci(Ci(x)^2-1)), x)

$\label{eq14}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{2 \ {{Ci \left({x}\right)}^{3}}}+{2 \ {Ci \left({x}\right)}}\right)}\ {\cos \left({x}\right)}\ {\sin \left({{{Ci \left({x}\right)}^{2}}+ 1}\right)}}+ \ \ \displaystyle {{\left(-{2 \ {{Ci \left({x}\right)}^{3}}}-{2 \ {Ci \left({x}\right)}}\right)}\ {\cos \left({x}\right)}\ {\cos \left({{{Ci \left({x}\right)}^{2}}- 1}\right)}}$

(14)

Type: Expression(Integer)

fricas

integrate(f, x)

$\label{eq15}1 \over{{Si \left({{{Ci \left({x}\right)}^{2}}+ 1}\right)}+{Ci \left({{{Ci \left({x}\right)}^{2}}- 1}\right)}}$

(15)

Type: Union(Expression(Integer),...)

fricas

f := D(log(Si(Ci(x)^2+1)+Ci(Ci(x)^2-1)), x)

$\label{eq16}{\left( \begin{array}{@{}l} \displaystyle {{\left({2 \ {{Ci \left({x}\right)}^{3}}}-{2 \ {Ci \left({x}\right)}}\right)}\ {\cos \left({x}\right)}\ {\sin \left({{{Ci \left({x}\right)}^{2}}+ 1}\right)}}+ \ \ \displaystyle {{\left({2 \ {{Ci \left({x}\right)}^{3}}}+{2 \ {Ci \left({x}\right)}}\right)}\ {\cos \left({x}\right)}\ {\cos \left({{{Ci \left({x}\right)}^{2}}- 1}\right)}}$

(16)

Type: Expression(Integer)

fricas

integrate(f, x)

$\label{eq17}\log \left({{Si \left({{{Ci \left({x}\right)}^{2}}+ 1}\right)}+{Ci \left({{{Ci \left({x}\right)}^{2}}- 1}\right)}}\right)$

(17)

Type: Union(Expression(Integer),...)

Alas, the following does not work

fricas

f := D(1/(1 + besselJ(m, x^2+a)), x)

$\label{eq18}{{x \ {besselJ \left({{m + 1}, \:{{{x}^{2}}+ a}}\right)}}-{x \ {besselJ \left({{m - 1}, \:{{{x}^{2}}+ a}}\right)}}}\over{{{besselJ \left({m , \:{{{x}^{2}}+ a}}\right)}^{2}}+{2 \ {besselJ \left({m , \:{{{x}^{2}}+ a}}\right)}}+ 1}$

(18)

Type: Expression(Integer)

fricas

integrate(f, x)

$\label{eq19}\int^{ \displaystyle x}{{{{\%A \ {besselJ \left({{m + 1}, \:{a +{{\%A}^{2}}}}\right)}}-{\%A \ {besselJ \left({{m - 1}, \:{a +{{\%A}^{2}}}}\right)}}}\over{{{besselJ \left({m , \:{a +{{\%A}^{2}}}}\right)}^{2}}+{2 \ {besselJ \left({m , \:{a +{{\%A}^{2}}}}\right)}}+ 1}}\ {d \%A}}$

(19)

Type: Union(Expression(Integer),...)

Risch algorithm wants to find elementary integral. For many functions integrals can be written only in terms of special function. FriCAS? now contains extensions which can handle some of them, see FriCASSpecialIntegration?.

For more elementaty examples see SandboxFriCASTimofeev1?, SandboxFriCASTimofeev37?, SandboxFriCASTimofeev5?, SandboxFriCASTimofeev9?.