How to calculate $\int \sqrt{(\cos{x})^2-a^2} \, dx$

Question

How to calculate: $$\int \sqrt{(\cos{x})^2-a^2} \, dx$$

Answer 1

$$\int\sqrt{\cos^2 x-a^2}\;dx =\frac{1}{k} \int \sqrt{1-k^2\sin^2x}\;dx$$ where $k=\frac{1}{\sqrt{1-a^2}}$ As this seems to come from a physical problem, introduce limits and look into elliptic integrals of the second kind.

Answer 2

In SWP (Scientific WorkPlace), with Local MAPLE kernel, I got the following evaluation

$$\begin{eqnarray*} I &:&=\int \sqrt{\cos ^{2}x-a^{2}}dx \\ &=&-\frac{\sqrt{\sin ^{2}x}}{\sin x}a^{2}\text{EllipticF}\left( \left( \cos x\right) \frac{\text{csgn}\left( a^{\ast }\right) }{a},\text{csgn}\left( a\right) a\right) \\ &&-\text{EllipticF}\left( \left( \cos x\right) \frac{\text{csgn}\left( a^{\ast }\right) }{a},\text{csgn}\left( a\right) a\right) \\ &&+\text{EllipticE}\left( \left( \cos x\right) \frac{\text{csgn}\left( a^{\ast }\right) }{a},\text{csgn}\left( a\right) a\right) F \end{eqnarray*}$$

where

$$F=\sqrt{\frac{-\cos ^{2}x+a^{2}}{a^{2}}}\sqrt{\cos ^{2}x-a^{2}}\text{csgn}\left( a^{\ast }\right) \frac{a}{-\cos ^{2}x+a^{2}}$$

As an example:

$$\begin{eqnarray*} \int \sqrt{\cos ^{2}x-2^{2}}dx &=&\frac{\sqrt{\sin ^{2}x}}{\sin x}3\text{EllipticF}\left( \frac{1}{2}\cos x,2\right) \\ &&+\text{EllipticE}\left( \frac{1}{2}\cos x,2\right) \frac{\sqrt{-\cos ^{2}x+4}}{\sqrt{\cos ^{2}x-4}} \end{eqnarray*}$$