Optimal. Leaf size=99 \[ -\frac{4 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |5\right )}{3 d} \]
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Rubi [A] time = 0.0993077, antiderivative size = 99, 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 = {2810, 2809} \[ -\frac{4 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |5\right )}{3 d} \]
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 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{2+3 \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)}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.769191, size = 194, normalized size = 1.96 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \left (3 F\left (\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )-5 \Pi \left (-\frac{2}{3};\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )\right )}{3 d \sqrt{-\cos (c+d x)} \sqrt{3 \cos (c+d x)+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.519, size = 159, normalized size = 1.6 \begin{align*} -{\frac{\sqrt{5}\sqrt{2}\sqrt{10} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( -1+\cos \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) } \left ({\it EllipticF} \left ({\frac{\sqrt{5} \left ( -1+\cos \left ( dx+c \right ) \right ) }{5\,\sin \left ( dx+c \right ) }},\sqrt{5} \right ) -2\,{\it EllipticPi} \left ( 1/5\,{\frac{\sqrt{5} \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},-5,\sqrt{5} \right ) \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{-\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{2+3\,\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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