Optimal. Leaf size=37 \[ \frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \]
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Rubi [A] time = 0.0566217, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {2771} \[ \frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2771
Rubi steps
\begin{align*} \int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0444338, size = 40, normalized size = 1.08 \[ \frac{2 \cot \left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.332, size = 46, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{-2\,a \left ( -1+\cos \left ( dx+c \right ) \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 [B] time = 1.57563, size = 111, normalized size = 3. \begin{align*} \frac{2 \,{\left (\sqrt{2} \sqrt{a} - \frac{\sqrt{2} \sqrt{a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84738, size = 113, normalized size = 3.05 \begin{align*} \frac{2 \, \sqrt{-a \cos \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) + 1\right )}}{d \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\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{- a \left (\cos{\left (c + d x \right )} - 1\right )}}{\cos ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.41281, size = 126, normalized size = 3.41 \begin{align*} -\frac{\sqrt{2}{\left (a^{2}{\left (\frac{\sqrt{2}}{\sqrt{a}{\left | a \right |}} - \frac{2}{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left | a \right |}}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) + \frac{\sqrt{2}{\left (\sqrt{2} a^{2} - a^{2}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{\sqrt{a}{\left | a \right |}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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