Integrals, Trig Substitutions with quartics: $\int{\frac{x}{{\left(x^2+2x+5\right)}^2}}dx$

Question

I am trying to evaluate using trig substitution: the indefinite integral

$$\int { \frac { x }{ { \left( x^{ 2 }+2x+5 \right) }^{ 2 } } } dx$$

I can easily do problems with smaller terms like $\frac { 1 }{ x\sqrt { x-5 } } $ but this problem is baffling me

Answer 1

Hint. By observing that $$ (x^2+2x+5)^2=\left[(x+1)^2+4 \right]^2 $$ the change of variable $$ x+1=2\tan u, \quad dx=2(\tan^2 u+1)du, $$ gives $$ \int\frac{x}{(x^2+2x+5)^2}\:dx=\frac1{8}\int \left(2\cos u \sin u-\cos^2 u \right)du $$ which might be easier to evaluate.