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partfrac

partfrac((4*x^3+12*x^...

Calculate

partfrac

f(p):=1/(p *(p^2 +1) ...

partfrac(f(z),z);

Calculate

partfrac

partfrac(1/(z^4+4),z);

Calculate

partfrac

partfrac((1.884*(z+0....

Calculate

partfrac

f(x):=(5*x^2-11*x+30)...

partfrac(f(x),x);

Calculate

partfrac

partfrac(1/(p*(1+p*L/...

Calculate

partfrac

partfrac(x-(1/a)-(1/b...

Calculate

partfrac

partfrac((z^2-1.833*z...

Calculate

partfrac

f(p):=1/(p *(p^2 +1) ...

partfrac(f(z),z);

Calculate

partfrac

partfrac(((3*x^2)/2 +...

Calculate

partfrac

Run Example
(%i1)simp:false;
(%o1)                                false
(%i2) radexpand:false;
(%o2)                                false
(%i3) 1/a, simp:false;
                                       1
(%o3)                                  -
                                       a
(%i4) 1/b, simp:false;
                                       1
(%o4)                                  -
                                       b
(%i5) 1/(a*b), simp:false;
                                       1
(%o5)                                 ---
                                      a b
(%i6) tellsimp(1/(a*b), ((1/a)*(1/b)));
(%o6)                         [/rule1, simpquot]
(%i7) 1/(a*b);
                                       1
(%o7)                                 ---
                                      a b
(%i8) disprule(all);
                                       1     1  1
(%t8)                        /rule1 : --- -> - (-)
                                      a b    a  b

(%o8)                                [%t8]
(%i9) 1/(a*b);
                                       1
(%o9)                                 ---
                                      a b
(%i10) properties("*");
(%o10)              [mirror symmetry, nary, rule, operator]
(%i11) rat(1/(a*b));
                                       1
(%o11)/R/                             ---
                                      a b
(%i12) map('ratsimp, 1/(a*b));
                                       1
(%o12)                                ---
                                      a b
(%i13) grind(1/(a*b));

1/(a*b)$
(%o13)                               done
(%i14) partfrac(1/(a*b), a);
                                     1  - 1
(%o14)                            + (- a   )
                                     b
(%i15) a^-1;
                                      - 1
(%o15)                               a
(%i16) 
Run Example
eo : (r1*r3-r2*r4)*ei/(r1*r2*(r3+r4)/rg+r3*r4*(r1+r2)/rg+(r1+r2)*(r3+r4));
                              ei (r1 r3 - r2 r4)
(%o1)       -------------------------------------------------------
            r1 r2 (r4 + r3)   (r2 + r1) r3 r4
            --------------- + --------------- + (r2 + r1) (r4 + r3)
                  rg                rg
(%i2) deo : ((r1+dr1)*(r3+dr3)-(r2+dr2)*(r4+dr4))*ei/((r1+dr1)*(r2+dr2)*((r3+dr3)+(r4+dr4))/rg+(r3+dr3)*(r4+dr4)*((r1+dr1)+(r2+dr2))/rg+((r1+dr1)+(r2+dr2))*((r3+dr3)+(r4+dr4)));
(%o2) (ei ((r1 + dr1) (r3 + dr3) - (r2 + dr2) (r4 + dr4)))
  (r1 + dr1) (r2 + dr2) (r4 + r3 + dr4 + dr3)
/(-------------------------------------------
                      rg
   (r2 + r1 + dr2 + dr1) (r3 + dr3) (r4 + dr4)
 + -------------------------------------------
                       rg
 + (r2 + r1 + dr2 + dr1) (r4 + r3 + dr4 + dr3))
(%i3) factor(deo/eo);
(%o3) ((r2 r4 + dr2 r4 - r1 r3 - dr1 r3 + dr4 r2 - dr3 r1 + dr2 dr4 - dr1 dr3)
 (r2 r4 rg + r1 r4 rg + r2 r3 rg + r1 r3 rg + r2 r3 r4 + r1 r3 r4 + r1 r2 r4
 + r1 r2 r3))/((r2 r4 - r1 r3) (r2 r4 rg + r1 r4 rg + dr2 r4 rg + dr1 r4 rg + r2 r3 rg + r1 r3 rg + dr2 r3 rg + dr1 r3 rg + dr4 r2 rg + dr3 r2 rg + dr4 r1 rg + dr3 r1 rg + dr2 dr4 rg + dr1 dr4 rg + dr2 dr3 rg + dr1 dr3 rg + r2 r3 r4
 + r1 r3 r4 + dr2 r3 r4 + dr1 r3 r4 + r1 r2 r4 + dr3 r2 r4 + dr1 r2 r4
 + dr3 r1 r4 + dr2 r1 r4 + dr2 dr3 r4 + dr1 dr3 r4 + dr1 dr2 r4 + r1 r2 r3
 + dr4 r2 r3 + dr1 r2 r3 + dr4 r1 r3 + dr2 r1 r3 + dr2 dr4 r3 + dr1 dr4 r3
 + dr1 dr2 r3 + dr4 r1 r2 + dr3 r1 r2 + dr3 dr4 r2 + dr1 dr4 r2 + dr1 dr3 r2
 + dr3 dr4 r1 + dr2 dr4 r1 + dr2 dr3 r1 + dr2 dr3 dr4 + dr1 dr3 dr4
 + dr1 dr2 dr4 + dr1 dr2
 dr3))
(%i4) partfrac(deo/eo,rg);
          2                             2
(%o4) ((r2  + (r1 + dr2) r2 + dr2 r1) r4
       2                      2                              2
 + ((r2  + (dr2 - dr1) r2 - r1  + (dr2 - dr1) r1) r3 + dr4 r2
                                                   2
 + ((dr4 - dr3) r1 + dr2 dr4 - dr1 dr3) r2 - dr3 r1  + (dr2 dr4 - dr1 dr3) r1)
                           2             2
 r4 + ((- r1 - dr1) r2 - r1  - dr1 r1) r3
          2                                                   2
 + (dr4 r2  + ((dr4 - dr3) r1 + dr2 dr4 - dr1 dr3) r2 - dr3 r1
                                   2                          2
 + (dr2 dr4 - dr1 dr3) r1) r3)/((r2  + (r1 + dr2 + dr1) r2) r4
       2                      2                                        2
 + ((r2  + (dr2 + dr1) r2 - r1  + (- dr2 - dr1) r1) r3 + (dr4 + dr3) r2
 + ((dr4 + dr3) r1 + (dr2 + dr1) dr4 + (dr2 + dr1) dr3) r2) r4
                2                       2
 + (- r1 r2 - r1  + (- dr2 - dr1) r1) r3
                                          2
 + ((- dr4 - dr3) r1 r2 + (- dr4 - dr3) r1
 + ((- dr2 - dr1) dr4 + (- dr2 - dr1) dr3) r1) r3)
                   3                                                2
 - (((dr3 + dr1) r2  + (2 dr3 r1 + (2 dr2 + dr1) dr3 + 2 dr1 dr2) r2
                  2                                          2
 + ((dr3 + dr2) r1  + ((3 dr2 + dr1) dr3 + dr1 dr2) r1 + (dr2  + dr1 dr2) dr3
          2                     2    2        2                         2
 + dr1 dr2 ) r2 + (dr2 dr3 + dr2 ) r1  + ((dr2  + dr1 dr2) dr3 + dr1 dr2 ) r1)
   3             3                                                2    2
 r4  + ((2 dr1 r2  + ((- dr3 - dr1) r1 - dr1 dr3 + 4 dr1 dr2 - dr1 ) r2
                      2
 + ((2 dr2 - 2 dr3) r1  + ((- dr2 - 3 dr1) dr3 + dr1 dr2) r1
                   2                 2      2                           3
 + (- dr1 dr2 - dr1 ) dr3 + 2 dr1 dr2  - dr1  dr2) r2 + (- dr3 - dr2) r1
                               2                2
 + ((- dr2 - 2 dr1) dr3 + 2 dr2  - 2 dr1 dr2) r1
                    2                 2      2
 + ((- dr1 dr2 - dr1 ) dr3 + 2 dr1 dr2  - dr1  dr2) r1) r3
                                     3                    2
 + ((2 dr3 + 2 dr1) dr4 + dr1 dr3) r2  + ((4 dr3 dr4 - dr3  - dr1 dr3) r1
                                                  2                   2
 + ((4 dr2 + 2 dr1) dr3 + 4 dr1 dr2) dr4 - dr1 dr3  + (2 dr1 dr2 - dr1 ) dr3)
   2                                2              2
 r2  + (((2 dr3 + 2 dr2) dr4 - 2 dr3  + dr2 dr3) r1
                                                               2
 + (((6 dr2 + 2 dr1) dr3 + 2 dr1 dr2) dr4 + (- dr2 - 3 dr1) dr3 ) r1
          2                             2                        2     2
 + ((2 dr2  + 2 dr1 dr2) dr3 + 2 dr1 dr2 ) dr4 + (- dr1 dr2 - dr1 ) dr3
           2      2                      2              3
 + (dr1 dr2  - dr1  dr2) dr3) r2 + (- dr3  - dr2 dr3) r1
                      2                           2       2
 + ((2 dr2 dr3 + 2 dr2 ) dr4 + (- dr2 - 2 dr1) dr3  + (dr2  - 2 dr1 dr2) dr3)
   2           2                             2                        2     2
 r1  + (((2 dr2  + 2 dr1 dr2) dr3 + 2 dr1 dr2 ) dr4 + (- dr1 dr2 - dr1 ) dr3
           2      2                 2                   3
 + (dr1 dr2  - dr1  dr2) dr3) r1) r4  + (((dr4 + dr1) r2
                                                              2    2
 + ((2 dr4 - 2 dr1) r1 + (2 dr2 + dr1) dr4 + 2 dr1 dr2 - 2 dr1 ) r2
                  2                                          2
 + ((dr4 + dr2) r1  + ((3 dr2 + dr1) dr4 - dr1 dr2) r1 + (dr2  + dr1 dr2) dr4
          2        2                   3                 2                2
 + dr1 dr2  - 2 dr1  dr2) r2 - 2 dr2 r1  + (dr2 dr4 + dr2  - 4 dr1 dr2) r1
        2                         2        2            2
 + ((dr2  + dr1 dr2) dr4 + dr1 dr2  - 2 dr1  dr2) r1) r3
                                    3
 + (((dr3 + 3 dr1) dr4 + dr1 dr3) r2  + (((dr3 - dr1) dr4 - 3 dr1 dr3) r1
                               2                          2         2
 + (2 dr2 dr3 + 6 dr1 dr2 - dr1 ) dr4 + (2 dr1 dr2 - 3 dr1 ) dr3) r2
                                    2
 + (((3 dr2 - dr3) dr4 + dr2 dr3) r1  + (((2 dr2 - 2 dr1) dr3 + 2 dr1 dr2) dr4
                            2      2                 2      2
 - 2 dr1 dr2 dr3) r1 + ((dr2  - dr1 ) dr3 + 3 dr1 dr2  - dr1  dr2) dr4
           2        2                                                  3
 + (dr1 dr2  - 3 dr1  dr2) dr3) r2 + ((- dr3 - dr2) dr4 - 3 dr2 dr3) r1
                        2                        2                     2
 + ((- 2 dr1 dr3 + 3 dr2  - 2 dr1 dr2) dr4 + (dr2  - 6 dr1 dr2) dr3) r1
         2      2                 2      2
 + (((dr2  - dr1 ) dr3 + 3 dr1 dr2  - dr1  dr2) dr4
           2        2                                    2                  3
 + (dr1 dr2  - 3 dr1  dr2) dr3) r1) r3 + ((dr3 + dr1) dr4  + dr1 dr3 dr4) r2
              2         2                         2
 + ((2 dr3 dr4  + (- dr3  - dr1 dr3) dr4 - dr1 dr3 ) r1
                                      2
 + ((2 dr2 + dr1) dr3 + 2 dr1 dr2) dr4
                    2               2           2    2    2
 + ((2 dr1 dr2 - dr1 ) dr3 - dr1 dr3 ) dr4 - dr1  dr3 ) r2
                    2                   2         2
 + (((dr3 + dr2) dr4  + (dr2 dr3 - 2 dr3 ) dr4) r1
                                     2                      2
 + (((3 dr2 + dr1) dr3 + dr1 dr2) dr4  + (- dr2 - 3 dr1) dr3  dr4
              2            2                         2     2
 - dr1 dr2 dr3 ) r1 + ((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
                    2     2           2      2                    2        2
 + ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
             2                         2    3
 r2 + ((- dr3  - dr2 dr3) dr4 - dr2 dr3 ) r1
                  2     2                       2       2
 + ((dr2 dr3 + dr2 ) dr4  + ((- dr2 - 2 dr1) dr3  + (dr2  - 2 dr1 dr2) dr3) dr4
                2    2         2                         2     2
 - 2 dr1 dr2 dr3 ) r1  + (((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
                    2     2           2      2                    2        2
 + ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
                                            2    2
 r1) r4 + (((- dr4 - dr1) r1 - dr1 dr4 - dr1 ) r2
              2                                                        2
 + (- 2 dr4 r1  + ((- dr2 - 3 dr1) dr4 - dr1 dr2) r1 + (- dr1 dr2 - dr1 ) dr4
      2                           3                                       2
 - dr1  dr2) r2 + (- dr4 - dr2) r1  + ((- dr2 - 2 dr1) dr4 - 2 dr1 dr2) r1
                    2           2            3
 + ((- dr1 dr2 - dr1 ) dr4 - dr1  dr2) r1) r3
        2              3          2
 + ((dr4  + dr1 dr4) r2  + ((2 dr4  + (- 2 dr3 - dr1) dr4 - 2 dr1 dr3) r1
                    2                                 2             2        2
 + (2 dr2 + dr1) dr4  + (- 2 dr1 dr3 + 2 dr1 dr2 - dr1 ) dr4 - 2 dr1  dr3) r2
        2                        2                     2
 + ((dr4  + (dr2 - 4 dr3) dr4) r1  + ((3 dr2 + dr1) dr4
                                                       2               2
 + (- 2 dr2 - 6 dr1) dr3 dr4 - 2 dr1 dr2 dr3) r1 + (dr2  + dr1 dr2) dr4
                        2               2      2                 2
 + ((- 2 dr1 dr2 - 2 dr1 ) dr3 + dr1 dr2  - dr1  dr2) dr4 - 2 dr1  dr2 dr3) r2
                                       3
 + ((- 2 dr3 - dr2) dr4 - 2 dr2 dr3) r1
           2                               2
 + (dr2 dr4  + ((- 2 dr2 - 4 dr1) dr3 + dr2  - 2 dr1 dr2) dr4 - 4 dr1 dr2 dr3)
   2        2               2                        2               2
 r1  + ((dr2  + dr1 dr2) dr4  + ((- 2 dr1 dr2 - 2 dr1 ) dr3 + dr1 dr2
      2                 2                2
 - dr1  dr2) dr4 - 2 dr1  dr2 dr3) r1) r3
                    2                  3
 + (((dr3 + dr1) dr4  + dr1 dr3 dr4) r2
              2         2                         2
 + ((2 dr3 dr4  + (- dr3  - dr1 dr3) dr4 - dr1 dr3 ) r1
                                      2
 + ((2 dr2 + dr1) dr3 + 2 dr1 dr2) dr4
                    2               2           2    2    2
 + ((2 dr1 dr2 - dr1 ) dr3 - dr1 dr3 ) dr4 - dr1  dr3 ) r2
                    2                   2         2
 + (((dr3 + dr2) dr4  + (dr2 dr3 - 2 dr3 ) dr4) r1
                                     2                      2
 + (((3 dr2 + dr1) dr3 + dr1 dr2) dr4  + (- dr2 - 3 dr1) dr3  dr4
              2            2                         2     2
 - dr1 dr2 dr3 ) r1 + ((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
                    2     2           2      2                    2        2
 + ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
             2                         2    3
 r2 + ((- dr3  - dr2 dr3) dr4 - dr2 dr3 ) r1
                  2     2                       2       2
 + ((dr2 dr3 + dr2 ) dr4  + ((- dr2 - 2 dr1) dr3  + (dr2  - 2 dr1 dr2) dr3) dr4
                2    2         2                         2     2
 - 2 dr1 dr2 dr3 ) r1  + (((dr2  + dr1 dr2) dr3 + dr1 dr2 ) dr4
                    2     2           2      2                    2        2
 + ((- dr1 dr2 - dr1 ) dr3  + (dr1 dr2  - dr1  dr2) dr3) dr4 - dr1  dr2 dr3 )
              2                          2       2                      2                                        2                                                                              2                       2                                          2
 r1) r3)/(((r2  + (r1 + dr2 + dr1) r2) r4  + ((r2  + (dr2 + dr1) r2 - r1  + (- dr2 - dr1) r1) r3 + (dr4 + dr3) r2  + ((dr4 + dr3) r1 + (dr2 + dr1) dr4 + (dr2 + dr1) dr3) r2) r4 + (- r1 r2 - r1  + (- dr2 - dr1) r1) r3  + ((- dr4 - dr3) r1 r2 + (- dr4 - dr3) r1  + ((- dr2 - dr1) dr4 + (- dr2 - dr1) dr3) r1) r3) (((r2 + r1 + dr2 + dr1) r4 + (r2 + r1 + dr2 + dr1) r3 + (dr4 + dr3) r2 + (dr4 + dr3) r1 + (dr2 + dr1) dr4 + (dr2 + dr1) dr3) rg + ((r2 + r1 + dr2 + dr1) r3 + (r1 + dr3 + dr1) r2 + (dr3 + dr2) r1 + (dr2 + dr1) dr3 + dr1 dr2) r4 + ((r1 + dr4 + dr1) r2 + (dr4 + dr2) r1 + (dr2 + dr1) dr4 + dr1 dr2) r3 + ((dr4 + dr3) r1 + (dr3 + dr1) dr4 + dr1 dr3) r2 + ((dr3 + dr2) dr4 + dr2 dr3) r1 + ((dr2 + dr1) dr3 + dr1 dr2) dr4 + dr1 dr2
 dr3))
(%i5) partfrac(eo,rg);
              2                          2    2
(%o5) (((ei r2  + ei r1 r2) r3 + ei r1 r2 ) r4
                       2    2            2        2                   2      2
 + ((- ei r1 r2 - ei r1 ) r3  + (ei r1 r2  - ei r1  r2) r3) r4 - ei r1  r2 r3 )
/(((r2 + r1) r4 + (r2 + r1) r3) (((r2 + r1) r4 + (r2 + r1) r3) rg + ((r2 + r1) r3 + r1 r2) r4 + r1 r2 r3))
       ei r2 r4 - ei r1 r3
 - ---------------------------
   (r2 + r1) r4 + (r2 + r1) r3
(%i6) partfrac((r1+dr1)*(r2+dr2)*((r3+dr3)+(r4+dr4))/rg+(r3+dr3)*(r4+dr4)*((r1+dr1)+(r2+dr2))/rg,rg);
(%o6) (((r2 + r1 + dr2 + dr1) r3 + (r1 + dr3 + dr1) r2 + (dr3 + dr2) r1
 + (dr2 + dr1) dr3 + dr1 dr2) r4 + ((r1 + dr4 + dr1) r2 + (dr4 + dr2) r1
 + (dr2 + dr1) dr4 + dr1 dr2) r3 + ((dr4 + dr3) r1 + (dr3 + dr1) dr4 + dr1 dr3)
 r2 + ((dr3 + dr2) dr4 + dr2 dr3) r1 + ((dr2 + dr1) dr3 + dr1 dr2) dr4
 + dr1 dr2 dr3)/rg
(%i7) expand((((r2 + r1 )* r3 + (r1)* r2 )* r4 + ((r1)* r2 )* r3) /rg);
                   r2 r3 r4   r1 r3 r4   r1 r2 r4   r1 r2 r3
(%o7)              -------- + -------- + -------- + --------
                      rg         rg         rg         rg
(%i8) ((((r2 + r1 )* r3 + (r1)* r2 )* r4 + ((r1)* r2 )* r3) /rg)/((r1+r2)*(r3+r4));
                     ((r2 + r1) r3 + r1 r2) r4 + r1 r2 r3
(%o8)                ------------------------------------
                            (r2 + r1) (r4 + r3) rg
(%i9) expand(r1*r2*(r3+r4)/rg+r3*r4*(r1+r2)/rg);
                   r2 r3 r4   r1 r3 r4   r1 r2 r4   r1 r2 r3
(%o9)              -------- + -------- + -------- + --------
                      rg         rg         rg         rg
(%i10) 
Run Example
f(z):=(5*z^2 -8*z +5)/((z-1)^2 *(z-i)^2 *(z+i)^2);
                                       2
                                    5 z  - 8 z + 5
(%o1)                 f(z) := --------------------------
                                     2        2        2
                              (z - 1)  (z - i)  (z + i)
(%i2) partfrac(f(z),z);
                3       2                               2
             5 i  + 11 i  + 15 i + 5                 5 i  + 8 i + 5
(%o2) ------------------------------------- + -----------------------------
          6       5       4      3                4      3      2         2
      (4 i  + 12 i  + 12 i  + 4 i ) (z + i)   (4 i  + 8 i  + 4 i ) (z + i)
             3       2                               2
          5 i  - 11 i  + 15 i - 5                 5 i  - 8 i + 5
 - ------------------------------------- + -----------------------------
       6       5       4      3                4      3      2         2
   (4 i  - 12 i  + 12 i  - 4 i ) (z - i)   (4 i  - 8 i  + 4 i ) (z - i)
                 2
              2 i  + 6                         2
 + ------------------------------ + ------------------------
     6      4      2                  4      2             2
   (i  - 3 i  + 3 i  - 1) (z - 1)   (i  - 2 i  + 1) (z - 1)
(%i3) ratsimp(%);
(%o3)             (%i4) 
[assume,declare,identfor,ilt,inf,invert,limit,matrix,numer,partfrac,ratsimp] [assume,declare,identfor,ilt,inf,invert,limit,matrix,numer,partfrac] [assume,declare,identfor,ilt,inf,invert,limit,matrix,partfrac] [atvalue,diff,factor,ilt,lambda,laplace,map,partfrac,solve] [atvalue,diff,ilt,lambda,laplace,map,partfrac,solve] [beta,gamma,partfrac] [clear_rules,disprule,false,grind,map,partfrac,properties,radexpand,rat,simp,tellsimpafter,true] [combine,cos,exp,partfrac,plot2d] [determinant,identfor,ilt,invert,matrix,partfrac] [diff,partfrac,sqrt] [disprule,facts,false,grind,map,partfrac,properties,radexpand,rat,simp,tellsimp] [disprule,false,grind,map,noeval,partfrac,properties,radexpand,rat,simp,tellsimp] [disprule,false,grind,map,partfrac,properties,radexpand,rat,simp,tellsimp] [disprule,false,grind,map,partfrac,properties,radexpand,rat,simp,tellsimpafter,true] [divide,num,partfrac,ratsimp] [echelon,integrate,kill,linsolve,matrix,partfrac,rank,submatrix] [echelon,integrate,kill,matrix,partfrac,rank,solve,submatrix] [echelon,integrate,kill,matrix,partfrac,rank,submatrix] [echelon,integrate,matrix,partfrac,rank,submatrix] [expand,factor,partfrac,ratsimp] [expand,factor,partfrac] [expand,partfrac,taylor] [factor,float,partfrac] [factor,partfrac] [grind,partfrac] [ilt,partfrac] [integrate,matrix,partfrac,rank,submatrix] [integrate,matrix,partfrac] [integrate,partfrac,plot2d] [integrate,partfrac] [kill,partfrac] [linsolve,partfrac] [matrix,partfrac,ratsimp] [partfrac,pi] [partfrac,plot2d] [partfrac,rat,sqrt,subst] [partfrac,ratsimp] [partfrac,sqrt] [partfrac,taylor] [partfrac]

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