<s>[INST] $\int \frac{2x}{9x^2+3}dx=?$ So this one seems very easy. And it really is (I guess). But I have an issue with this one. I solved it this way: 
\begin{align}
& \int \frac{2x}{9x^2+3}dx=\frac{1}{6} \int \frac{x}{3x^2+1}dx=\frac{1}{6} \left( x\frac{\arctan(\sqrt3x)}{\sqrt{3}}-\int 1\cdot\frac{\arctan(\sqrt3x)}{\sqrt3} dx\right) \\[10pt]
= {} & \frac{1}{6} \left( x\frac{\arctan(\sqrt3x)}{\sqrt{3}}-x\frac{\arctan(\sqrt3x)}{\sqrt{3}}+\frac{\arctan(\sqrt3x)}{\sqrt3}\right)=\frac{\arctan(\sqrt3x)}{\sqrt3}+C
\end{align}
But at the same time, the textbook says, that it is equal to this:
$$
\frac{1}{9}\log(3x^2+1)+C
$$
Did I do something wrong? It would be quite hard for me to believ that these to functions are the same. 
Thanks for the responses! 
 [/INST] I'd have written $u = 9x^2+ 3$ so $du = 18x\,dx$ and $2x\,dx = \dfrac{du} 9$.  Then you have
$$
\int \frac{2x}{9x^2+3} \, dx = \frac 1 9 \int \frac{du} u = \frac 1 9 \log|u|+C = \frac 1 9 \log ( 9x^2+3) + C.
$$
This is the same as
$$
\frac 1 9 \log ( 3( 3x^2+1)) + C = \frac 1 9 \left( \log(3x^2+1) + \log 3 \right) + C = \frac 1 9 \log(3x^2+1) + (\text{a different } C).
$$
Your initial integration by parts says
$$
\frac{1}{6} \int \frac{x}{3x^2+1}dx=\frac{1}{6} \left( x\frac{\arctan(\sqrt3x)}{\sqrt{3}}-\int 1\cdot\frac{\arctan(\sqrt3x)}{\sqrt3} dx\right).
$$
That's alright as far as it goes.  I haven't attempted to figure out just what you're doing after that, but let's recall how to antidifferentiate the arctangent function:
\begin{align}
& \int \arctan x\,dx = \overbrace{\int u\,dx = xu - \int x\,du}^\text{integration by parts} \\[10pt]
= {} & x\arctan x - \int \frac{x\,dx}{1+x^2} = x\arctan x - \int \frac{dw/2}{w} \\[10pt]
= {} & x \arctan x - \frac 1 2 \log|w|+C \\[10pt]
= {} & x \arctan x - \frac 1 2 \log(1+x^2) + C.
\end{align}
So I suspect the first term in the result just above this line (suitably adapted with your $\sqrt3$, etc.) will cancel out the first term in your integration by parts and then the result will ultimately be the same.
Generally I wouldn't integrate by parts when a simple substitution will work, unless there were some special reason to do that.
</s>