Optimal. Leaf size=95 \[ -\frac{3 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d} \]
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Rubi [A] time = 0.100234, antiderivative size = 95, 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, 2808} \[ -\frac{3 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2810
Rule 2808
Rubi steps
\begin{align*} \int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{3+2 \cos (c+d x)}} \, dx &=\frac{\sqrt{-\cos (c+d x)} \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{3+2 \cos (c+d x)}} \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{3 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{3+2 \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)}}{d}\\ \end{align*}
Mathematica [A] time = 0.50089, size = 119, normalized size = 1.25 \[ -\frac{2 \sqrt{-\cos (c+d x)} \sqrt{2 \cos (c+d x)+3} \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )+2 \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )\right )}{\sqrt{5} d \sqrt{(3 \cos (c+d x)+\cos (2 (c+d x))+1) \sec ^4\left (\frac{1}{2} (c+d x)\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.385, size = 168, normalized size = 1.8 \begin{align*}{\frac{{\frac{i}{5}}\sqrt{5}\sqrt{10}\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d \left ( -1+\cos \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) } \left ({\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 ) -2\,{\it EllipticPi} \left ({\frac{i/5 \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},5,i\sqrt{5} \right ) \right ) \sqrt{-\cos \left ( dx+c \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 ) }}}} \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{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}}, 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 \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{\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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