Integrand size = 25, antiderivative size = 56 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \]
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Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2893, 2892} \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \]
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Rule 2892
Rule 2893
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \, dx}{\sqrt {\cos (c+d x)}} \\ & = -\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(56)=112\).
Time = 1.71 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2-3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(51)=102\).
Time = 7.70 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.91
method | result | size |
default | \(\frac {2 \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2-3 \cos \left (d x +c \right )}\, \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \left (-2+3 \cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}\) | \(107\) |
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 - 3 \cos {\left (c + d x \right )}} \sqrt {\cos {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \]
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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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