Optimal. Leaf size=121 \[ \frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{6 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.23856, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4222, 2762, 21, 2772, 2771} \[ \frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{6 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 4222
Rule 2762
Rule 21
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}-\frac{1}{5} \left (2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{9 a}{2}-\frac{9}{2} a \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx\\ &=\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}+\frac{1}{5} \left (9 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{6 a^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}+\frac{1}{5} \left (6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{12 a^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}+\frac{6 a^2 \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{5 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.186384, size = 62, normalized size = 0.51 \[ \frac{2 a (3 \cos (c+d x)+3 \cos (2 (c+d x))+4) \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.382, size = 73, normalized size = 0.6 \begin{align*} -{\frac{2\,a \left ( 6\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\cos \left ( dx+c \right ) -1 \right ) \cos \left ( dx+c \right ) }{5\,d\sin \left ( dx+c \right ) } \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{7}{2}}}\sqrt{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 [B] time = 1.62323, size = 293, normalized size = 2.42 \begin{align*} \frac{4 \,{\left (\frac{5 \, \sqrt{2} a^{\frac{3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sqrt{2} a^{\frac{3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{7 \, \sqrt{2} a^{\frac{3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{2 \, \sqrt{2} a^{\frac{3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{5 \, 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{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 = 1.62508, size = 197, normalized size = 1.63 \begin{align*} \frac{2 \,{\left (6 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{5 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )} \sqrt{\cos \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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