Optimal. Leaf size=62 \[ -\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]
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Rubi [A] time = 0.05566, antiderivative size = 62, 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} \[ -\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]
Antiderivative was successfully verified.
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Rule 2815
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{-3+2 \cos (c+d x)}} \, dx &=-\frac{2 \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{-3+2 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right ) \sqrt{-\tan ^2(c+d x)}}{\sqrt{5} d}\\ \end{align*}
Mathematica [B] time = 0.633724, size = 160, normalized size = 2.58 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{-\cot ^2\left (\frac{1}{2} (c+d x)\right )} \cot (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{-(2 \cos (c+d x)-3) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{2 \cos (c+d x)-3}{\cos (c+d x)-1}}}{\sqrt{3}}\right )|\frac{6}{5}\right )}{\sqrt{5} d (-\cos (c+d x))^{3/2} \sqrt{2 \cos (c+d x)-3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.256, size = 98, normalized size = 1.6 \begin{align*} 2\,{\frac{\sqrt{2}\sqrt{-3+2\,\cos \left ( dx+c \right ) }}{d\sqrt{-\cos \left ( dx+c \right ) }}{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{-2\,{\frac{-3+2\,\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{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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