Optimal. Leaf size=161 \[ \frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}+\frac{7 \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{4 d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.301612, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2778, 2983, 2982, 2782, 206, 2775, 207} \[ \frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}+\frac{7 \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{4 d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2778
Rule 2983
Rule 2982
Rule 2782
Rule 206
Rule 2775
Rule 207
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{1-\cos (c+d x)}} \, dx &=\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{1}{4} \int \frac{\sqrt{\cos (c+d x)} (3+\cos (c+d x))}{\sqrt{1-\cos (c+d x)}} \, dx\\ &=\frac{\sqrt{\cos (c+d x)} \sin (c+d x)}{4 d \sqrt{1-\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1-\cos (c+d x)}}-\frac{1}{4} \int \frac{-\frac{1}{2}-\frac{7}{2} \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)}{4 d \sqrt{1-\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1-\cos (c+d x)}}-\frac{7}{8} \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)}{4 d \sqrt{1-\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1-\cos (c+d x)}}-\frac{7 \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 )}{4 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{7 \tanh ^{-1}\left (\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right )}{4 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)}{4 d \sqrt{1-\cos (c+d x)}}+\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt{1-\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.41225, size = 255, normalized size = 1.58 \[ -\frac{i e^{-2 i (c+d x)} \left (-1+e^{i (c+d x)}\right ) \sqrt{\cos (c+d x)} \left (7 \sqrt{2} e^{2 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )-16 e^{2 i (c+d x)} \tanh ^{-1}\left (\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )+\sqrt{2} \left (\sqrt{1+e^{2 i (c+d x)}} \left (2 e^{i (c+d x)}+2 e^{2 i (c+d x)}+e^{3 i (c+d x)}+1\right )+7 e^{2 i (c+d x)} \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )\right )}{8 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{1-\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.329, size = 194, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( 2\,\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\cos \left ( dx+c \right ) -4\,{\it Artanh} \left ( 1/2\,{\sqrt{2}{\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \right ) \sqrt{2}+\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+7\,{\it Artanh} \left ( \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{5}{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{5}{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 [A] time = 2.00554, size = 653, normalized size = 4.06 \begin{align*} \frac{4 \, \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 (2 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{-\cos \left (d x + c\right ) + 1} \sqrt{\cos \left (d x + c\right )} + 7 \, \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 ) - 7 \, \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 )}{8 \, 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.0876, size = 219, normalized size = 1.36 \begin{align*} -\frac{\sqrt{2}{\left (7 \, \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 \,{\left ({\left (-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{\frac{3}{2}} + 2 \, \sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}} + 8 \, \log \left (\sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 1\right ) - 8 \, \log \left (-\sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 1\right )\right )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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