Finding integrals of the form $\int\frac{1}{x^2(x^{2009}+1)^{ \frac{2008}{2009}}}dx$

Question

I faced two similar integrals today.

They are

$$\int\frac{1}{x^2(x^{2009}+1)^{ \frac{2008}{2009}}}dx$$

and $$\int\frac{1}{x^2(x^{2009}+1)^{ \frac{1}{2009}}}dx$$

No trigonometric substitution is working here.I tried almost all.What to do?

Moreover binomial expansion was perhaps possible but that would be really very ugly...

Edit: Oh and yes another one too which is nearly similar but I could'nt solve.

$$\int\frac{1}{{e^x}({e}^{2008x}+1)^{ \frac{2007}{2008}}}dx$$

Answer 1

Let $$I = \int\frac{1}{x^2(x^{2009}+1)^{\frac{2008}{2008}}}dx = \int\frac{1}{x^2\cdot x^{2008}(1+x^{-2009})^{\frac{2008}{2009}}}dx$$

Now Put $1+x^{-2009} = t\;,$ Then $\displaystyle x^{-2010}dx = -\frac{1}{2009}dt$

So $$I = -\frac{1}{2009}\int t^{-\frac{2008}{2009}}dt$$

For the second one:: $$I = \int\frac{1}{e^x(e^{2008x}+1)^{\frac{2007}{2008}}}dx\;,$$ Put $e^x=t\;,$ and $e^xdx=dt$

So we get $$I = \int\frac{1}{t^2(t^{2008}+1)^{\frac{2007}{2008}}}dt$$