From: drhuang57@gmail.com   
      
   On Tuesday, 2 May 2017 16:42:07 UTC+10, Nasser M. Abbasi  wrote:   
   > 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 ArcSi   
   [x])])+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))))+(   
   *log(((-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)*co   
   (u)+(-1)*(-1)^(1/   
   >    2))))+((-1)*(-1)^(1/2)*dilog((2*sin(u))/(sin(u)+((-1)^(1/2)   
   cos(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)   
   cos(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   
      
   Your answer is too complicated. mathhandbook give a simple one:   
   http://www.mathhandbook.com/input/?guess=integrate%28cos%28x%29%   
   Fsin%28x%29*x%29&inp=cos%28x%29%2Fsin%28x%29*x   
      
   and show graph.   
      
   reference math handbook   
   www.mathHandbook.com   
      
   --- SoupGate-Win32 v1.05   
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