Optimal. Leaf size=75 \[ -\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d} \]
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Rubi [A] time = 0.0525886, antiderivative size = 75, 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} \[ -\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d} \]
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{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{-3-2 \cos (c+d x)}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right ) \sqrt{1-\sec (c+d x)} \sqrt{1+\sec (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.294639, size = 117, normalized size = 1.56 \[ \frac{2 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x) (2 \cos (c+d x)+3) \sec ^4\left (\frac{1}{2} (c+d x)\right )} \left (F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )+2 \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )\right )}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.405, size = 164, normalized size = 2.2 \begin{align*}{\frac{\sqrt{10}\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) -3 \right ) \cos \left ( dx+c \right ) } \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},-1,i/5\sqrt{5} \right ) \right ) \sqrt{{\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 ) }}}\sqrt{-3-2\,\cos \left ( dx+c \right ) }\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{\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 ) + 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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