![\" \light \int_0^1\frac{x^3 + 1}{x^4 + 4x + 1}dx\"](\"https:\/\/walkingrandomly.com\/wp-content\/ql-cache\/quicklatex.com-e8201a99e32ba9873612c294a8d9316e_l3.png\" "\\"Rendered")
<\/p>\nHow good is your symbolic integral calculus? Do you think that you can do better than Mathematica? Let’s see – try and evaluate the following (there is a hint in the comments if you get stuck).<\/p>\n
<\/p>\n
My integration skills are a little rusty but I found the solution, log(6)\/4, without too much difficulty (I have picked up the habit of writing log(x) when I mean ln(x) from using computer algebra packages too much) . Let’s see how Mathematica handles the same integration. Plugging the following command into version 6<\/p>\n
Integrate[(x^3 + 1)\/(x^4 + 4*x + 1), {x, 0, 1}]<\/p>\n
gives a solution of<\/p>\n
(RootSum[1 + 4*#1 + #1^4 & , Log[1 – #1]\/(1 + #1^3) & ] -RootSum[1 + 4*#1 + #1^4 & , Log[-#1]\/(1 + #1^3) & ] +
\nRootSum[1 + 4*#1 + #1^4 & ,(Log[1 – #1]*#1^3)\/(1 + #1^3) & ] -RootSum[1 + 4*#1 + #1^4 & , (Log[-#1]*#1^3)\/(1 + #1^3) & ])\/4<\/p>\n
Ugh! Applying the FullSimplify command doesn’t help so it seems that this is the best that Mathematica can do at the moment. If you evaluate this expression numerically then it agrees with the symbolic result but I think you would agree that Mathematica has not done a very good job here.<\/p>\n
I found this integral while looking through the changelog<\/a> of the latest version of Maxima<\/a> – an open source mathematics package. If you try and evaluate it in pre-5.14 versions of Maxima then it will appear to hang (actually it will return a result eventually if you leave it long enough and have enough memory but it makes the Mathematica result look positively pretty). This has been fixed in version 5.14 and now issuing the command<\/p>\n integrate ((x^3 + 1)\/(x^4 + 4*x + 1),x,0,1);<\/p>\n