# [Maxima] integrate((x^2+1)/(x^4+1),x,0,1) can't be simplified

Juan Pablo Romero Bernal jromerobernal at gmail.com
Mon Apr 28 01:50:13 CDT 2008

```Hi,

2008/4/27 cunzhang <apzc2529 at gmail.com>:
> hi,sorry for my unclear expression. I mean
>
>
>                     (%i1) integrate((x^2+1)/(x^4+1),x,0,1);
>                            sqrt(2) + 2         2 - sqrt(2)
>                       atan(-----------)   atan(-----------)
>                              sqrt(2)             sqrt(2)
>                       ----------------- + -----------------
>                            sqrt(2)             sqrt(2)
>  should be simplified to sqrt(2)/4*%pi, say,
>
>                       sqrt(2) + 2         2 - sqrt(2)
>                  atan(-----------)   atan(-----------)
>                         sqrt(2)             sqrt(2)      %pi
>                  ----------------- + ----------------- = ----
>                       sqrt(2)             sqrt(2)         3/2
>                                                          2
>
>  is exact.
>  So how  can Maxima do it?
>

Yes. With the package atrig1, you can to apply several simplification
rules to inverse trigonometric functions. In this case, i've added two
rules to the Maxima simplifier:

(%i1) tellsimpafter(atan(sqrt(2)-1),3*%pi/8)\$

(%i2) tellsimpafter(atan(sqrt(2)+1),%pi/8)\$

(%i3) integrate((x^2+1)/(x^4+1),x,0,1)\$

(%i4) expand(%);
%pi
(%o4)                              ---------
2 sqrt(2)

If you want, you can to add this rules into the atrig1, package, in the file:

\$MAXIMA_INSTALLATION_DIR/share/trigonometry/atrig1.mac

Bye,

--
Juan Pablo Romero Bernal