Optimal. Leaf size=77 \[ \frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (-\frac{1}{2};\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]
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Rubi [A] time = 0.0532438, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2808} \[ \frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (-\frac{1}{2};\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]
Antiderivative was successfully verified.
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Rule 2808
Rubi steps
\begin{align*} \int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-3+2 \cos (c+d x)}} \, dx &=\frac{3 \cot (c+d x) \Pi \left (-\frac{1}{2};\sin ^{-1}\left (\frac{\sqrt{-3+2 \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)}}{\sqrt{5} d}\\ \end{align*}
Mathematica [C] time = 0.149898, size = 140, normalized size = 1.82 \[ \frac{2 i \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{2 \cos (c+d x)-3} \left (F\left (i \sinh ^{-1}\left (\sqrt{5} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )-2 \Pi \left (\frac{1}{5};i \sinh ^{-1}\left (\sqrt{5} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )\right )}{\sqrt{5} d \sqrt{-\cos (c+d x)} \sqrt{\frac{3-2 \cos (c+d x)}{\cos (c+d x)+1}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.365, size = 152, normalized size = 2. \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d \left ( -1+\cos \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) } \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},-1,i\sqrt{5} \right ) \right ) \sqrt{-\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{-3+2\,\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{\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}}, 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{\sqrt{- \cos{\left (c + d x \right )}}}{\sqrt{2 \cos{\left (c + d x \right )} - 3}}\, 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{\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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