From: nma@12000.org   
      
   Hello;   
      
   Why can't Fricas integrate asin(x)/x dx ?  Since there is standard   
   substitution u=asin(x) which converts the integrand to   
      
           u*cos(u)/sin(u) du   
      
   which Fricas can integrate with no problem?   
      
   Illustration using Rubi:   
      
   In[88]:= Int[ArcSin[x]/x,x]   
   Out[88]= -(1/2) I ArcSin[x]^2+ArcSin[x] Log[1-E^(2 I ArcSin[x])]-1/2 I   
   PolyLog[2,E^(2 I ArcSin[x])]   
      
   And now using the transformed version of the integrand:   
      
   In[91]:= Int[u*Cos[u]/Sin[u],u]   
   Out[91]= -((I u^2)/2)+u Log[1-E^(2 I u)]-1/2 I PolyLog[2,E^(2 I u)]   
      
   Convert back to x   
      
   In[92]:= %/.u->ArcSin[x]//Simplify   
   Out[92]= -(1/2) I (ArcSin[x] (ArcSin[x]+2 I Log[1-E^(2 I ArcSin[   
   ])])+PolyLog[2,E^(2 I ArcSin[x])])   
      
   we see [92] is same as [88].   
      
   But Fricas does not like   
      
     ii:=integrate(asin(x)/x,x)   
      
   but it can do the transformed version   
      
   ii:=integrate(u*cos(u)/sin(u),u)   
      
   (u*log((2*sin(u))/(sin(u)+((-1)*(-1)^(1/2)*cos(u)+(-1)*(-1)^(1/2))))+(u*log(   
      (2*sin(u))/(sin(u)+((-1)^(1/2)*cos(u)+(-1)^(1/2))))+(u*log((2   
   (-1)^(1/2)*cos(   
      u)+2*(-1)^(1/2))/(sin(u)+((-1)^(1/2)*cos(u)+(-1)^(1/2))))+(u*   
   og(((-2)*(-1)^(   
      1/2)*cos(u)+(-2)*(-1)^(1/2))/(sin(u)+((-1)*(-1)^(1/2)*cos(u)+   
   -1)*(-1)^(1/2))   
      ))+((-1)^(1/2)*dilog((2*sin(u))/(sin(u)+((-1)*(-1)^(1/2)*cos(   
   )+(-1)*(-1)^(1/   
      2))))+((-1)*(-1)^(1/2)*dilog((2*sin(u))/(sin(u)+((-1)^(1/2)*c   
   s(u)+(-1)^(1/2)   
      )))+((-1)*(-1)^(1/2)*dilog((2*(-1)^(1/2)*cos(u)+2*(-1)^(1/2))   
   (sin(u)+((-1)^(   
      1/2)*cos(u)+(-1)^(1/2))))+(-1)^(1/2)*dilog(((-2)*(-1)^(1/2)*c   
   s(u)+(-2)*(-1)^   
      (1/2))/(sin(u)+((-1)*(-1)^(1/2)*cos(u)+(-1)*(-1)^(1/2)))))))))))/2   
      
   Just wondering why, that is all. I am using   
      
     FriCAS Computer Algebra System   
       Version: FriCAS 1.3.0   
      
   Thank you,   
   --Nasser   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)