Optimal. Leaf size=161 \[ \frac{46 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{92 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.286009, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2762, 2980, 2772, 2771} \[ \frac{46 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{92 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2762
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{1}{7} (2 a) \int \frac{\left (-\frac{15 a}{2}-\frac{11}{2} a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{7} \left (23 a^2\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{46 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (46 a^2\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{46 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{92 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 5.2457, size = 74, normalized size = 0.46 \[ \frac{a^2 (93 \cos (c+d x)+23 \cos (2 (c+d x))+23 \cos (3 (c+d x))+29) \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{21 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.36, size = 77, normalized size = 0.5 \begin{align*} -{\frac{2\,{a}^{2} \left ( 46\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-23\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-11\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-9\,\cos \left ( dx+c \right ) -3 \right ) }{21\,d\sin \left ( dx+c \right ) }\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62213, size = 328, normalized size = 2.04 \begin{align*} \frac{8 \,{\left (\frac{21 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{56 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{63 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{36 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{8 \, \sqrt{2} a^{\frac{5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{21 \, d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{9}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{9}{2}}{\left (\frac{2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96982, size = 242, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (46 \, a^{2} \cos \left (d x + c\right )^{3} + 23 \, a^{2} \cos \left (d x + c\right )^{2} + 12 \, a^{2} \cos \left (d x + c\right ) + 3 \, a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{21 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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