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changevar

Run Example
(%i1)'integrate((x+(x^2+1)^(1/2))^(-n),x,1,+inf);
                           inf
                          /
                          [             1
(%o1)                     I    ------------------- dx
                          ]           2          n
                          /    (sqrt(x  + 1) + x)
                           1
(%i2) changevar(%,y-(sqrt(x^2+1)+x),y,x);
(%o2)             
Run Example
integrate(x/(x^3+1),x);
                                        2 x - 1
                      2            atan(-------)
                 log(x  - x + 1)        sqrt(3)    log(x + 1)
(%o1)            --------------- + ------------- - ----------
                        6             sqrt(3)          3
(%i2) ratsimp (diff(integrate(x/(x^3+1),x),x));
                                      x
(%o2)                               ------
                                     3
                                    x  + 1
(%i3) integrate(1/((x-3)^4+1/2),x);
                               /
                               [      1
(%o3)                          I ------------ dx
                               ]        4   1
                               / (x - 3)  + -
                                            2
(%i4) changevar(%,x-3-y,y,x);
(%o4)             (%i5) 
Run Example
integrate((sqrt(x)-sqrt(x+1))/(sqrt(x)+sqrt(x+1)),x);
                          /
                          [ sqrt(x) - sqrt(x + 1)
(%o1)                     I --------------------- dx
                          ] sqrt(x + 1) + sqrt(x)
                          /
(%i2) f(x):='integrate((sqrt(x)-sqrt(x+1))/(sqrt(x)+sqrt(x+1)),x);
                              /
                              [ sqrt(x) - sqrt(x + 1)
(%o2)                 f(x) := I --------------------- dx
                              ] sqrt(x) + sqrt(x + 1)
                              /
(%i3)  assume(y>
0);
(%o3)                               [y > 0]
(%i4) changevar(f(x),y^2=x/(x+1),y,x);
                            /
                            [          y
(%o4)                     2 I ------------------- dy
                            ]  4      3
                            / y  + 2 y  - 2 y - 1
(%i5)  %,nouns;
(%o5)             

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