Optimal. Leaf size=75 \[ -\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{1}{3};\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right )|\frac{1}{5}\right )}{3 \sqrt{5} d} \]
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Rubi [A] time = 0.0590285, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2809} \[ -\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{1}{3};\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right )|\frac{1}{5}\right )}{3 \sqrt{5} d} \]
Antiderivative was successfully verified.
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Rule 2809
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-2+3 \cos (c+d x)}} \, dx &=-\frac{4 \cot (c+d x) \Pi \left (\frac{1}{3};\sin ^{-1}\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}\\ \end{align*}
Mathematica [A] time = 0.6622, size = 142, normalized size = 1.89 \[ -\frac{4 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{3 \cos (c+d x)-2}{\cos (c+d x)+1}} \left (F\left (\sin ^{-1}\left (\sqrt{5} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{1}{5}\right )+2 \Pi \left (-\frac{1}{5};-\sin ^{-1}\left (\sqrt{5} \tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{1}{5}\right )\right )}{\sqrt{5} d \sqrt{\cos (c+d x)} \sqrt{3 \cos (c+d x)-2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.428, size = 132, normalized size = 1.8 \begin{align*} -2\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d\sqrt{-2+3\,\cos \left ( dx+c \right ) } \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{\cos \left ( dx+c \right ) }} \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},\sqrt{5} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},-1,\sqrt{5} \right ) \right ) \sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\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{3 \, \cos \left (d x + c\right ) - 2}}\,{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{3 \, \cos \left (d x + c\right ) - 2}}, 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{3 \cos{\left (c + d x \right )} - 2}}\, 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{3 \, \cos \left (d x + c\right ) - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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