Integrand size = 25, antiderivative size = 101 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d \sqrt {-\cos (c+d x)}} \]
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Time = 0.13 (sec) , antiderivative size = 101, 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 = {2889, 2888} \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),5\right )}{3 d \sqrt {-\cos (c+d x)}} \]
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Rule 2888
Rule 2889
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos (c+d x)} \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)}} \\ & = -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {-2-3 \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)}}{3 d \sqrt {-\cos (c+d x)}} \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {-\frac {(2+3 \cos (c+d x))^2}{(1+\cos (c+d x))^2}} \left (\operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )-2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {-\frac {2+3 \cos (c+d x)}{1+\cos (c+d x)}}} \]
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Time = 6.59 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.48
method | result | size |
default | \(\frac {\left (F\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right )-2 \Pi \left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, -5, \sqrt {5}\right )\right ) \sqrt {2}\, \sqrt {10}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-2-3 \cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 d \sqrt {\cos \left (d x +c \right )}\, \left (2+3 \cos \left (d x +c \right )\right )}\) | \(149\) |
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\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2}} \,d x } \]
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\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {- 3 \cos {\left (c + d x \right )} - 2}}\, dx \]
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\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2}} \,d x } \]
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\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \]
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