for example this problem http://www.geocities.com/coolageofempire/trig.JPG

after solving it using trib substitution, my answer has square root inside the natural log but mathematica gives an answer same as mine without the square root so what is the problem here?

I completed the square in the denominator and used the substitution [imath]\tan\theta =x+\frac{3}{2}[/imath]. This gave me

 

[math]\int \frac{x}{4x^2+12x+13}dx=-\frac{1}{4}\ln\left(\frac{2}{\sqrt{4x^2+12x+13}}\right)-\frac{3}{8}\tan^{-1}\left(x+\frac{3}{2}\right)+c[/math]

 

[math]=\frac{1}{4}\ln\left(\frac{\sqrt{4x^2+12x+13}}{2}\right)-\frac{3}{8}\tan^{-1}\left(x+\frac{3}{2}\right)+c[/math]

 

I differentiated this, and it gave me the correct original integrand. Maybe mathematica removed the power of a half or something from the log.