$\begin{array}{ll}{\int }_{0}^{2\pi }\frac{{cos}^{2}x}{{\Delta }^{3}}dx\hfill & =\frac{2{cos}^{2}\gamma }{{p}^{2}-{q}^{2}}\left[{sinh}^{2}\alpha {cosh}^{2}\beta \frac{{I}_{1}}{{p}^{4}}+{cosh}^{2}\alpha {sinh}^{2}\beta \frac{{I}_{2}}{{q}^{4}}\right]\hfill \\ +\frac{2{cosh}^{2}\alpha {cosh}^{2}\beta {sin}^{2}\gamma }{{p}^{2}{q}^{2}}\left(\frac{{I}_{1}}{{p}^{2}}-\frac{{I}_{2}}{{q}^{2}}\right)\hfill \end{array}$

7.

$\begin{array}{ll}{\int }_{0}^{2\pi }\frac{cosxsinx}{{\Delta }^{3}}dx\hfill & =sinh\alpha cosh\alpha sinh\beta cosh\beta sin\gamma cos\gamma \hfill \\ ×\left[\frac{2}{{p}^{2}-{q}^{2}}\left(\frac{{I}_{1}}{{p}^{4}}+\frac{{I}_{2}}{{q}^{4}}\right)-\frac{2}{{p}^{2}{q}^{2}}\left(\frac{{I}_{1}}{{p}^{2}}-\frac{{I}_{2}}{{q}^{2}}\right)\right]\hfill \end{array}$

3.679(3)*

1.

${\int }_{0}^{\pi /2}\frac{dx}{\sqrt{a{cos}^{2}x+b{sin}^{2}x}}={R}_{F}\left(0,a,b\right)$

DLMF (19.23.1)

[Rea > 0, Reb > 0]

2.

${\int }_{0}^{\pi /2}\sqrt{a{cos}^{2}x+b{sin}^{2}x}\text{d}x=2{R}_{G}\left(0,a,b\right)$

DLMF (19.23.2)

[Rea > 0, Reb > 0]

3.

${\int }_{0}^{}$

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