Table of Integrals, Series, and Products, 8th Edition

3.¹² $\begin{array}{l}\begin{array}{c}{\displaystyle \int \frac{{\text{tan}}^{4}x\text{d}x}{\sqrt{\left(1-p^2{\text{sin}}^{2}x\right)\left(1-q^2{\text{sin}}^{2}x\right)}}}\hfill \\ =\hfill & \frac{1}{3{\left(1-q^2\right)}^2{\left(1-p^2\right)}^{\frac{3}{2}}}\times \left[2\left(2-p^2-q^2\right)E\left(\alpha ,\sqrt{\frac{q^2-p^2}{1-p^2}}\right)-\left(1-q^2\right)F(\alpha ,\sqrt{\frac{q^2-p^2}{1-p^2}})\right]\hfill \\ +\frac{2p^2+q^2-3+{\text{sin}}^{2}x\left(4-3p^2-2p^2+p^2{q}^2\right)}{3\left(1-p^2\right){\left(1-q^2\right)}^2}\frac{\text{sin}x}{{\text{cos}}^{3}x}\sqrt{\frac{1-q^2{\text{sin}}^{2}x}{1-p^2{\text{sin}}^{2}x}}\hfill \end{array}\hfill \\ \left[0<p^2<q^2<1,0<x\le \frac{\pi }{2}\right]\hfill \end{array}$BY (284.07)

4. 

$\begin{array}{l}\begin{array}{c}{\displaystyle \int \frac{{\text{sin}}^{2}x\text{d}x}{\sqrt{\left(1-p^2{\text{sin}}^{2}x\right)}{\left(1-q^2{\text{sin}}^{2}x\right)}^3}}\hfill \\ =\hfill & \frac{\sqrt{1-p^2}}{\left(1-q^2\right)\left(q^2-p^2\right)}E\left(\alpha ,\sqrt{\frac{q^2-p^2}{1-p^2}}\right)-\frac{1}{\left(q^2-p^2\right)\sqrt{1-p^2}}F\left(\alpha ,\sqrt{\frac{q^2-p^2}{1-p^2}}\right)\hfill \\ -\frac{\text{sin}x\text{cos}x}{\left(1-q^2\right)\sqrt{(1}}\hfill \end{array}\hfill \end{array}$