Integrand size = 27, antiderivative size = 54 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1+\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)}} \]
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Time = 0.12 (sec) , antiderivative size = 54, 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 = {2893, 2892} \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{\cos (c+d x)+1}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {-\cos (c+d x)}} \]
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Rule 2892
Rule 2893
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {2+3 \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. \(150\) vs. \(2(54)=108\).
Time = 0.46 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.78 \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=-\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs. \(2(49)=98\).
Time = 7.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.06
method | result | size |
default | \(\frac {\left (1+\cos \left (d x +c \right )\right ) F\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {5}}{5 d \sqrt {2+3 \cos \left (d x +c \right )}\, \sqrt {-\cos \left (d x +c \right )}}\) | \(111\) |
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\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {- \cos {\left (c + d x \right )}} \sqrt {3 \cos {\left (c + d x \right )} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-\cos \left (d x + c\right )} \sqrt {3 \, \cos \left (d x + c\right ) + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-\cos (c+d x)} \sqrt {2+3 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {-\cos \left (c+d\,x\right )}\,\sqrt {3\,\cos \left (c+d\,x\right )+2}} \,d x \]
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