Optimal. Leaf size=80 \[ \frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt{-\cos (c+d x)}} \]
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Rubi [A] time = 0.104798, antiderivative size = 80, 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 = {2817, 2815} \[ \frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt{-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2817
Rule 2815
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{3+2 \cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{3+2 \cos (c+d x)}} \, dx}{\sqrt{-\cos (c+d x)}}\\ &=\frac{2 \cos ^{\frac{3}{2}}(c+d x) \csc (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 \sqrt{-\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.599877, size = 154, normalized size = 1.92 \[ -\frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{-\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{(2 \cos (c+d x)+3) \csc ^2\left (\frac{1}{2} (c+d x)\right )}}{\sqrt{6}}\right )\right |6\right )}{d \sqrt{-\cos (c+d x)} \sqrt{2 \cos (c+d x)+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.481, size = 127, normalized size = 1.6 \begin{align*}{\frac{-{\frac{i}{5}}\sqrt{5}\sqrt{2}\sqrt{10} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d \left ( -1+\cos \left ( dx+c \right ) \right ) }{\it EllipticF} \left ({\frac{{\frac{i}{5}} \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\frac{1}{\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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