Optimal. Leaf size=118 \[ -\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \]
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Rubi [A] time = 0.172602, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2772, 2771} \[ -\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}-\frac{4}{5} \int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}-\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{8}{15} \int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}-\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.147521, size = 62, normalized size = 0.53 \[ \frac{2 (-4 \cos (c+d x)+4 \cos (2 (c+d x))+7) \cot \left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \cos (c+d x)}}{15 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.34, size = 66, normalized size = 0.6 \begin{align*} -{\frac{\sqrt{2} \left ( 8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4\,\cos \left ( dx+c \right ) +3 \right ) \sin \left ( dx+c \right ) }{15\,d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{-2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54642, size = 298, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (7 \, \sqrt{2} \sqrt{a} - \frac{17 \, \sqrt{2} \sqrt{a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{25 \, \sqrt{2} \sqrt{a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{15 \, \sqrt{2} \sqrt{a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}{\left (\frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87124, size = 169, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (8 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 3\right )} \sqrt{-a \cos \left (d x + c\right ) + a}}{15 \, d \cos \left (d x + c\right )^{\frac{5}{2}} \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 = 2.20359, size = 219, normalized size = 1.86 \begin{align*} -\frac{\sqrt{2}{\left (2 \, a^{2}{\left (\frac{4 \, \sqrt{2}}{\sqrt{a}{\left | a \right |}} - \frac{15 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} + 20 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a + 12 \, a^{2}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{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 (7 \, \sqrt{2} a^{2} - 8 \, 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 )}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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