Integrand size = 27, antiderivative size = 34 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d} \]
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Time = 0.07 (sec) , antiderivative size = 34, 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 = {2892} \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d} \]
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Rule 2892
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(34)=68\).
Time = 0.52 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.26 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=-\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2-3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(33)=66\).
Time = 6.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.21
method | result | size |
default | \(\frac {2 \left (1+\cos \left (d x +c \right )\right ) F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2-3 \cos \left (d x +c \right )}}{d \sqrt {-\cos \left (d x +c \right )}\, \left (-2+3 \cos \left (d x +c \right )\right )}\) | \(109\) |
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {-3 \, \cos \left (d x + c\right ) + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {2 - 3 \cos {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {-3 \, \cos \left (d x + c\right ) + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {-3 \, \cos \left (d x + c\right ) + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {2-3\,\cos \left (c+d\,x\right )}} \,d x \]
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