Integrand size = 25, antiderivative size = 99 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {1}{3},\arcsin \left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right ),\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}} \]
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Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2889, 2888} \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (\frac {1}{3},\arcsin \left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right ),\frac {1}{5}\right )}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}} \]
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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 {1}{3},\arcsin \left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right ),\frac {1}{5}\right ) \sqrt {-1+\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(201\) vs. \(2(99)=198\).
Time = 1.62 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2-3 \cos (c+d x)}} \, dx=\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\left ((-2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )\right )} \csc (c+d x) \left (3 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {(2-3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),\frac {4}{5}\right )-\operatorname {EllipticPi}\left (\frac {2}{3},\arcsin \left (\frac {1}{2} \sqrt {(2-3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),\frac {4}{5}\right )\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{3 \sqrt {5} d \sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \]
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Time = 6.71 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.31
method | result | size |
default | \(-\frac {2 \left (F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )-2 \Pi \left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \sqrt {5}\right )\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2-3 \cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right )}\, \left (-2+3 \cos \left (d x +c \right )\right )}\) | \(130\) |
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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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\[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2-3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {2 - 3 \cos {\left (c + d x \right )}}}\, dx \]
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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. \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2-3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {2-3\,\cos \left (c+d\,x\right )}} \,d x \]
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