Integrand size = 27, antiderivative size = 99 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx=-\frac {4 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {2+3 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d} \]
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Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2889, 2888} \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx=-\frac {4 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {3 \cos (c+d x)+2}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),5\right )}{3 d} \]
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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 \sqrt {-\cos (c+d x)} \sqrt {\cos (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} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.96 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx=\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \left (3 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right )-5 \operatorname {EllipticPi}\left (-\frac {2}{3},\arcsin \left (\frac {1}{2} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right )\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{3 d \sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \]
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Time = 6.91 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.40
method | result | size |
default | \(-\frac {\sqrt {-\cos \left (d x +c \right )}\, \left (F\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right )-2 \Pi \left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, -5, \sqrt {5}\right )\right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \left (1+\sec \left (d x +c \right )\right ) \sqrt {5}}{5 d \sqrt {2+3 \cos \left (d x +c \right )}}\) | \(139\) |
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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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