Optimal. Leaf size=39 \[ \frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)-a}}\right )}{d} \]
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Rubi [A] time = 0.049979, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2773, 204} \[ \frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)-a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2773
Rule 204
Rubi steps
\begin{align*} \int \csc (c+d x) \sqrt{-a+a \sin (c+d x)} \, dx &=-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{-a+a \sin (c+d x)}}\right )}{d}\\ &=\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{-a+a \sin (c+d x)}}\right )}{d}\\ \end{align*}
Mathematica [B] time = 0.0766714, size = 96, normalized size = 2.46 \[ \frac{\sqrt{a (\sin (c+d x)-1)} \left (\log \left (-\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.382, size = 70, normalized size = 1.8 \begin{align*} 2\,{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) \sqrt{-a \left ( 1+\sin \left ( dx+c \right ) \right ) }\sqrt{a}}{\cos \left ( dx+c \right ) \sqrt{a\sin \left ( dx+c \right ) -a}d}\arctan \left ({\frac{\sqrt{-a \left ( 1+\sin \left ( dx+c \right ) \right ) }}{\sqrt{a}}} \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 \sqrt{a \sin \left (d x + c\right ) - a} \csc \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49424, size = 587, normalized size = 15.05 \begin{align*} \left [\frac{\sqrt{-a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \,{\left (\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt{a \sin \left (d x + c\right ) - a} \sqrt{-a} - 9 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right )}{2 \, d}, -\frac{\sqrt{a} \arctan \left (\frac{\sqrt{a \sin \left (d x + c\right ) - a}{\left (\sin \left (d x + c\right ) + 2\right )}}{2 \, \sqrt{a} \cos \left (d x + c\right )}\right )}{d}\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 \sqrt{a \left (\sin{\left (c + d x \right )} - 1\right )} \csc{\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 = 2.13176, size = 269, normalized size = 6.9 \begin{align*} -\frac{2 \, \sqrt{a} \arctan \left (-\frac{\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a}}{\sqrt{a}}\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) - \sqrt{-a} \log \left ({\left | -\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a} \right |}\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) -{\left (2 \, \sqrt{a} \arctan \left (\frac{\sqrt{2} \sqrt{-a} - \sqrt{-a}}{\sqrt{a}}\right ) - \sqrt{-a} \log \left ({\left | \sqrt{2} \sqrt{-a} - \sqrt{-a} \right |}\right )\right )} \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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