How to solve this Indefinite Integral step by step?

Question

Integrate

$$ \int \frac{\sqrt{(x^2+1)^5}}{x^6} \; dx $$ Trying this question but unable to solve. Wolfram does it with hyperbolic substitution, I have been asked to solve this integral without hyperbolic substitution.

Answer 1

Let $x=\tan u$ therefore $$\int \frac{\sqrt{(x^2+1)^5}}{x^6} \; dx =\int\dfrac{1}{\cos^7 u}\dfrac{\cos^6 u}{\sin^6 u}du=\int\dfrac{\cos udu}{\sin^6 u(1-\sin^2 u)}$$let $w=\sin u$ therefore $$\int\dfrac{\cos udu}{\sin^6 u(1-\sin^2 u)}=\int\dfrac{dw}{w^6(1-w^2)}=\int\dfrac{0.5}{1-w}+\dfrac{0.5}{1+w}+\dfrac{1}{w^6}+\dfrac{1}{w^4}+\dfrac{1}{w^2}dw\\=0.5\ln|\dfrac{w+1}{w-1}|-\dfrac{1}{5w^5}-\dfrac{1}{3w^3}-\dfrac{1}{w}+C$$

Answer 2

HINT \begin{align*} \frac{\sqrt{(x^{2}+1)^{5}}}{x^{6}} = \frac{1}{x}\frac{\sqrt{(x^{2}+1)^{5}}}{x^{5}} = \frac{1}{x}\sqrt{\frac{(x^{2}+1)^{5}}{x^{10}}} = \frac{1}{x}\sqrt{\left(\frac{x^{2}+1}{x^{2}}\right)^{5}} = \frac{1}{x}\sqrt{\left(1 + \frac{1}{x^{2}}\right)^{5}} \end{align*}