The value of $\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$ is

$$\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) \mathrm{d} x=$$

If, $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{e}}|\mathrm{f}(\theta)|+\mathrm{c}$ (where c is a constant of integration), then the ordered pair $(\lambda,|f(\theta)|)$ is equal to

If $\quad \int(2 x+4) \sqrt{x-1} \mathrm{~d} x=\mathrm{a}(x-1)^{\frac{5}{2}}+\mathrm{b}(x-1)^{\frac{3}{2}}+\mathrm{c}$, (where c is a constant of integration), then the value of $a+b$ is