Optimal. Leaf size=60 \[ -\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d} \]
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Rubi [A] time = 0.0556197, antiderivative size = 60, 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 = {2815} \[ -\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2815
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx &=-\frac{2 \cot (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{-3-2 \cos (c+d x)}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right ) \sqrt{-\tan ^2(c+d x)}}{d}\\ \end{align*}
Mathematica [B] time = 0.440684, size = 155, normalized size = 2.58 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{3}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right )|\frac{6}{5}\right )}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.429, size = 128, normalized size = 2.1 \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 ) }{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \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 ) }{\frac{1}{\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{1}{\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 )^{2} + 3 \, \cos \left (d x + c\right )}, 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{1}{\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{1}{\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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