Optimal. Leaf size=118 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}+\frac{\tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.213549, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2778, 2982, 2782, 206, 2775, 207} \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}+\frac{\tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2778
Rule 2982
Rule 2782
Rule 206
Rule 2775
Rule 207
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{1-\cos (c+d x)}} \, dx &=\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{1-\cos (c+d x)}}+\frac{1}{2} \int \frac{1+\cos (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx\\ &=\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{1-\cos (c+d x)}}-\frac{1}{2} \int \frac{\sqrt{1-\cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx+\int \frac{1}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx\\ &=\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{1-\cos (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d}+\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{d \sqrt{1-\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.158419, size = 227, normalized size = 1.92 \[ -\frac{i e^{-i (c+d x)} \left (-1+e^{i (c+d x)}\right ) \sqrt{\cos (c+d x)} \left (\sqrt{2} e^{i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )-4 e^{i (c+d x)} \tanh ^{-1}\left (\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )+\sqrt{2} \left (\sqrt{1+e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )+e^{i (c+d x)} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )\right )}{2 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{1-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.31, size = 161, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\cos \left ( dx+c \right ) -{\it Artanh} \left ({\frac{\sqrt{2}}{2}{\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \right ) \sqrt{2}+{\it Artanh} \left ( \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) +\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2-2\,\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{\cos \left (d x + c\right )^{\frac{3}{2}}}{\sqrt{-\cos \left (d x + c\right ) + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02748, size = 617, normalized size = 5.23 \begin{align*} \frac{\sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} \cos \left (d x + c\right ) + \sqrt{2}\right )} \sqrt{-\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} -{\left (3 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 2 \,{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt{-\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} + \log \left (\frac{2 \,{\left (\sqrt{-\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} + \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - \log \left (\frac{2 \,{\left (\sqrt{-\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} - \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right )}{2 \, d \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{\cos ^{\frac{3}{2}}{\left (c + d x \right )}}{\sqrt{1 - \cos{\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 = 2.14402, size = 190, normalized size = 1.61 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{2} \log \left (\frac{\sqrt{2} - \sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}{\sqrt{2} + \sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}\right ) - \frac{4 \, \sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 2 \, \log \left (\sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 1\right ) - 2 \, \log \left (-\sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 1\right )\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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