finding value of indefinite integration of $\frac{y^7-y^5+y^3-y}{y^{10}+1}dy$ finding value of indefinite integration  $\displaystyle \int\frac{y^7-y^5+y^3-y}{y^{10}+1}dy$
$\displaystyle \int\frac{y^5(y^2-1)+y(y^2-1)}{y^{10}+1}dy = \int\frac{(y^5+y)(y^2-1)}{y^{10}+1}dy = \int\frac{(y^5+y)(y^2+1-2)}{y^{10}+1}dy$
$\displaystyle =\int\frac{y^5+y}{y^8-y^6+y^4-y^2+1}-2\int\frac{y^5+y}{y^{10}+1}dy$
i wan,t be able to proceed after that, could some help me with this

Hint:   We can write our integral as, $$I = \int \frac{y^7-y^5+y^3-y}{y^{10}+1} dy = \int \frac{y(y^6-y^4 + y^2-1)}{y^{10}+1} dy = \int \frac{y(y^6-y^4+y^2-1)}{(y^2+1)(y^8-y^6+y^4-y^2+1)} dy$$ We can use partial fractions and get $$I = \frac{1}{5}\int \frac{4y^7-3y^5 +2y^3-y}{y^8-y^6+y^4-y^2+1} dy -\frac{4}{5} \int \frac{y}{y^2+1} dy =I_1-I_2$$ I hope you can take it from here.
