calculus - Compute: $int_{0}^{1}frac{x^4+1}{x^6+1} dx$

I'm trying to compute:

$$\int_{0}^{1}\frac{x^4+1}{x^6+1}dx.$$

I tried to change $x^4$ into $t^2$ or $t$, but it didn't work for me.

Any suggestions?

Thanks!

Answer

Edited Here is a much simpler version of the previous answer.

$$\int_0^1 \frac{x^4+1}{x^6+1}dx =\int_0^1 \frac{x^4-x^2+1}{x^6+1}dx+ \int_0^1 \frac{x^2}{x^6+1}dx$$

After canceling the first fraction, and subbing $y=x^3$ in the second we get:

$$\int_0^1 \frac{x^4+1}{x^6+1}dx =\int_0^1 \frac{1}{x^2+1}dx+ \frac{1}{3}\int_0^1 \frac{1}{y^2+1}dy = \frac{\pi}{4}+\frac{\pi}{12}=\frac{\pi}{3} \,.$$

P.S. Thanks to Zarrax for pointing the stupid mistakes I did...