Let $a$ be a non zero real number. Evaluate the integral $\int \frac{-7x}{x^{4}-a^{4}}dx$ I hit a wall on this question. Below are my steps
$$\int \frac{-7x}{x^{4}-a^{4}}dx=-7\int \frac{x}{x^{4}-a^{4}}dx$$
Let $u=\frac{x^2}{2}, dx = \frac{du}{x}, x^{4}=4u^{2}.$
$$-7\int \frac{1}{4u^{2}-a^{4}}du=-7\int \frac{1}{(2u+a^2)(2u-a^2)}du$$
Use partial fraction decomposition,
$$\frac{1}{(2u+a^2)(2u-a^2)}=\frac{A}{2u+a^{2}}+\frac{B}{2u-a^{2}}.$$
Solve for $A$ and $B$:
$$\begin{cases}
A=\frac{1}{-2a^{2}}
\\
B=\frac{1}{2a^{2}}
\end{cases}$$
Now $$\int \frac{1}{(2u+a^2)(2u-a^2)}du=\int \frac{1}{-2a^2(2u+a^{2})}+\int \frac{1}{2a^{2}(2u-a^{2})}$$
Factoring out $a$ yields
$$\frac{7}{2a^{2}}(\int \frac{1}{2u+a^{2}}-\int \frac{1}{2u-a^{2}})$$
Evaluate the integral and substitute $u=\frac{x^{2}}{2}$ back.
My final answer is $$\frac{7}{2a^{2}}(\log(x^2+a^2)-\log(x^2-a^2)).$$
Feedback says my answer is wrong. Where did I mess up?

$$I=\int\frac{-7x}{x^4-a^4}dx=-7\int\frac{x}{x^4-a^4}dx$$
now for ease lets let $u=x^2\Rightarrow dx=\frac{du}{2x}$ and $b=a^2$ so:
$$I=-\frac72\int\frac{du}{u^2-b^2}$$
now let $u=bv\Rightarrow du=b\,dv$ so:
$$I=-\frac{7b}{2}\int\frac{dv}{b^2(v^2-1)}=-\frac{7}{2b}\int\frac{dv}{v^2-1}$$
now this is a standard integral that you can solve with PFD or using a substitution
