Optimal. Leaf size=35 \[ \frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.0424697, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2771} \[ \frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2771
Rubi steps
\begin{align*} \int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0436007, size = 39, normalized size = 1.11 \[ \frac{2 \sqrt{1-\cos (c+d x)} \cot \left (\frac{1}{2} (c+d x)\right )}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.279, size = 45, normalized size = 1.3 \begin{align*} -{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{2-2\,\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 [B] time = 1.53573, size = 101, normalized size = 2.89 \begin{align*} \frac{2 \,{\left (\sqrt{2} - \frac{\sqrt{2} \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 = 1.85173, size = 111, normalized size = 3.17 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt{-\cos \left (d x + c\right ) + 1}}{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{1 - \cos{\left (c + d x \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 [A] time = 2.12484, size = 84, normalized size = 2.4 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{2}{\left (\sqrt{2} - 1\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) +{\left (\sqrt{2} - \frac{2}{\sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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