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FriCASIntegration

Edit detail for FriCASIntegration revision 2 of 13

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Editor: test1 Time: 2015/01/08 16:08:02 GMT+0
Note:

changed:
-For more elementaty examples see SandboxFriCASTimofeev1, SandboxFriCASTimofeev37,
For more elementaty examples see FriCASTimofeev1, SandboxFriCASTimofeev37,

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 FriCASTimofeev1?, SandboxFriCASTimofeev37?, SandboxFriCASTimofeev5?, SandboxFriCASTimofeev9?.