Integrand size = 27, antiderivative size = 60 \[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=-\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \]
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Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2894} \[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right )}{d} \]
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Rule 2894
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right ) \sqrt {-\tan ^2(c+d x)}}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(155\) vs. \(2(60)=120\).
Time = 0.44 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3+2 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{3}} \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}}\right ),\frac {6}{5}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{\sqrt {5} d \sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (57 ) = 114\).
Time = 6.84 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.97
method | result | size |
default | \(\frac {\left (1+\cos \left (d x +c \right )\right ) F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-3-2 \cos \left (d x +c \right )}}{5 d \sqrt {-\cos \left (d x +c \right )}\, \left (3+2 \cos \left (d x +c \right )\right )}\) | \(118\) |
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\[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {-2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {- 2 \cos {\left (c + d x \right )} - 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {-2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {-2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {-2\,\cos \left (c+d\,x\right )-3}} \,d x \]
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