Optimal. Leaf size=173 \[ \frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.40089, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2779, 2984, 12, 2782, 208} \[ \frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 2779
Rule 2984
Rule 12
Rule 2782
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{\int \frac{a+4 a \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx}{5 a}\\ &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}-\frac{2 \int \frac{-\frac{13 a^2}{2}-a^2 \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx}{15 a^2}\\ &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}+\frac{4 \int \frac{15 a^3}{4 \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \, dx}{15 a^3}\\ &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}+\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \, dx\\ &=\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{2 a^2-a x^2} \, dx,x,\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{d}\\ &=-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right )}{\sqrt{a} d}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.653393, size = 218, normalized size = 1.26 \[ \frac{e^{-\frac{5}{2} i (c+d x)} \sin \left (\frac{1}{2} (c+d x)\right ) \left (2 \sqrt{1+e^{2 i (c+d x)}} \left (15 e^{i (c+d x)}+40 e^{2 i (c+d x)}+40 e^{3 i (c+d x)}+15 e^{4 i (c+d x)}+13 e^{5 i (c+d x)}+13\right )-15 \sqrt{2} \left (1+e^{2 i (c+d x)}\right )^3 \tanh ^{-1}\left (\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )\right )}{60 d \sqrt{1+e^{2 i (c+d x)}} \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.366, size = 305, normalized size = 1.8 \begin{align*}{\frac{\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{15\,d \left ( -1+\cos \left ( dx+c \right ) \right ) ^{3} \left ( 1+\cos \left ( dx+c \right ) \right ) ^{3}} \left ( 15\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}{\it Artanh} \left ( 1/2\,{\sqrt{2}{\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \right ) +45\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}{\it Artanh} \left ( 1/2\,{\sqrt{2}{\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \right ) +45\,\cos \left ( dx+c \right ) \sqrt{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}{\it Artanh} \left ( 1/2\,{\sqrt{2}{\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \right ) +15\,\sqrt{2} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{7/2}{\it Artanh} \left ( 1/2\,{\sqrt{2}{\frac{1}{\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \right ) -26\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-6\,\cos \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{-2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37675, size = 485, normalized size = 2.8 \begin{align*} \frac{15 \, \sqrt{2} \sqrt{a} \cos \left (d x + c\right )^{3} \log \left (-\frac{\frac{2 \, \sqrt{2} \sqrt{-a \cos \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a}} -{\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 ) + 4 \,{\left (13 \, \cos \left (d x + c\right )^{3} + 14 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 3\right )} \sqrt{-a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{30 \, a d \cos \left (d x + c\right )^{3} \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 = 2.25681, size = 184, normalized size = 1.06 \begin{align*} \frac{\sqrt{2} a{\left (\frac{15 \, \arctan \left (\frac{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}{\left | a \right |}} + \frac{2 \,{\left (15 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} + 10 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a + 12 \, a^{2}\right )}}{{\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 )}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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