Integrand size = 27, antiderivative size = 75 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=-\frac {3 \cot (c+d x) \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \]
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Time = 0.06 (sec) , antiderivative size = 75, 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 = {2887} \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=-\frac {3 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right )}{d} \]
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Rule 2887
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cot (c+d x) \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=-\frac {2 i \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5}}\right ),-5\right )-2 \operatorname {EllipticPi}\left (5,i \text {arcsinh}\left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5}}\right ),-5\right )\right ) \sqrt {\cos (c+d x) (3+2 \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )}}{d \sqrt {-3-2 \cos (c+d x)} \sqrt {-\cos (c+d x)}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (68 ) = 136\).
Time = 5.94 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.92
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\cos \left (d x +c \right )}\, \sqrt {-3-2 \cos \left (d x +c \right )}\, \left (F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {i \sqrt {5}}{5}\right )-2 \Pi \left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \frac {i \sqrt {5}}{5}\right )\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (1+\sec \left (d x +c \right )\right )}{5 d \left (3+2 \cos \left (d x +c \right )\right )}\) | \(144\) |
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\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]
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\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {- \cos {\left (c + d x \right )}}}{\sqrt {- 2 \cos {\left (c + d x \right )} - 3}}\, dx \]
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\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]
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\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-2 \, \cos \left (d x + c\right ) - 3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=\int \frac {\sqrt {-\cos \left (c+d\,x\right )}}{\sqrt {-2\,\cos \left (c+d\,x\right )-3}} \,d x \]
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