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> f:=(x+1)/(x^3+x^2-2*x);
 

(x+1)/(x^3+x^2-2*x) 

> F:=int(f,x);
 

-1/2*ln(x)-1/6*ln(x+2)+2/3*ln(x-1) 

> f:=1/(x^3+1);
 

1/(x^3+1) 

> F:=int(f,x);
 

1/3*ln(x+1)-1/6*ln(x^2-x+1)+1/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2)) 

> f:=(x-11)/(x^3+1);
 

(x-11)/(x^3+1) 

> F:=int(f,x);
 

-4*ln(x+1)+2*ln(x^2-x+1)-10/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2)) 

> f:=(x^3+4*x^2+2*x+1)/(x^2+3);
 

(x^3+4*x^2+2*x+1)/(x^2+3) 

> F:=int(f,x);
 

1/2*x^2+4*x-1/2*ln(x^2+3)-11/3*3^(1/2)*arctan(1/3*x*3^(1/2)) 

> f:=1/(x+2*sqrt(x-1));
 

1/(x+2*(x-1)^(1/2)) 

> F:=int(f,x);
 

-2/(x-2)+ln(x-2)+ln((x-1)^(1/2)+1)+1/((x-1)^(1/2)-1)+1/((x-1)^(1/2)+1)-ln((x-1)^(1/2)-1)
-2/(x-2)+ln(x-2)+ln((x-1)^(1/2)+1)+1/((x-1)^(1/2)-1)+1/((x-1)^(1/2)+1)-ln((x-1)^(1/2)-1) 

> F:=expand(F);
 

-2/(x-2)+ln(x-2)+ln((x-1)^(1/2)+1)+1/((x-1)^(1/2)-1)+1/((x-1)^(1/2)+1)-ln((x-1)^(1/2)-1)
-2/(x-2)+ln(x-2)+ln((x-1)^(1/2)+1)+1/((x-1)^(1/2)-1)+1/((x-1)^(1/2)+1)-ln((x-1)^(1/2)-1) 

> f:=1/sqrt(x^2-1);
 

1/(x^2-1)^(1/2) 

> F:=int(f,x);
 

ln(x+(x^2-1)^(1/2)) 

> f:=1/(x^2*sqrt(1+x^2));
 

1/(x^2*(1+x^2)^(1/2)) 

> F:=int(f,x);
 

-(1+x^2)^(1/2)/x 

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