Optimal. Leaf size=99 \[ -\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{1}{3};\sin ^{-1}\left (\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|\frac{1}{5}\right )}{3 \sqrt{5} d \sqrt{-\cos (c+d x)}} \]
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Rubi [A] time = 0.116352, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2810, 2809} \[ -\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{1}{3};\sin ^{-1}\left (\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|\frac{1}{5}\right )}{3 \sqrt{5} d \sqrt{-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2810
Rule 2809
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{2-3 \cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{2-3 \cos (c+d x)}} \, dx}{\sqrt{-\cos (c+d x)}}\\ &=-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (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 \sqrt{-\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.69056, size = 147, normalized size = 1.48 \[ -\frac{4 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{-\frac{(2-3 \cos (c+d x))^2}{(\cos (c+d x)+1)^2}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left (F\left (\left .\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )\right |5\right )+2 \Pi \left (-1;\left .-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )\right |5\right )\right )}{d \sqrt{2-3 \cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{\frac{2-3 \cos (c+d x)}{\cos (c+d x)+1}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.399, size = 144, normalized size = 1.5 \begin{align*} 2\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}\sqrt{2-3\,\cos \left ( dx+c \right ) }}{d \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-5\,\cos \left ( dx+c \right ) +2 \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{-3 \, \cos \left (d x + c\right ) + 2} \sqrt{\cos \left (d x + c\right )}}{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{2 - 3 \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{\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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