Optimal. Leaf size=82 \[ \frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d \sqrt{-\cos (c+d x)}} \]
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Rubi [A] time = 0.105928, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2817, 2815} \[ \frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d \sqrt{-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2817
Rule 2815
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \frac{1}{\sqrt{3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx}{\sqrt{-\cos (c+d x)}}\\ &=\frac{2 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right )|-\frac{1}{5}\right ) \sqrt{-\tan ^2(c+d x)}}{\sqrt{5} d \sqrt{-\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.468222, size = 146, normalized size = 1.78 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{(3-2 \cos (c+d x)) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}}{\sqrt{3}}\right )\right |6\right )}{d \sqrt{3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.279, size = 107, normalized size = 1.3 \begin{align*}{\frac{{\frac{2\,i}{5}}\sqrt{5}\sqrt{2}}{d}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{3-2\,\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}{\frac{1}{\sqrt{-\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cos \left (d x + c\right )} \sqrt{-2 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-\cos \left (d x + c\right )} \sqrt{-2 \, \cos \left (d x + c\right ) + 3}}{2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \cos{\left (c + d x \right )}} \sqrt{3 - 2 \cos{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\cos \left (d x + c\right )} \sqrt{-2 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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