Optimal. Leaf size=156 \[ \frac{64 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
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Rubi [A] time = 0.299475, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {4264, 3812, 3809, 3804} \[ \frac{64 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 4264
Rule 3812
Rule 3809
Rule 3804
Rubi steps
\begin{align*} \int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} \left (5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} \left (8 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{16 a^2 \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{21} \left (32 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{16 a^2 \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{2 \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.255969, size = 74, normalized size = 0.47 \[ \frac{a^2 \sqrt{\cos (c+d x)} (101 \cos (c+d x)+24 \cos (2 (c+d x))+3 \cos (3 (c+d x))+208) \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)}}{42 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.168, size = 85, normalized size = 0.5 \begin{align*} -{\frac{2\,{a}^{2} \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+9\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+11\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+23\,\cos \left ( dx+c \right ) -46 \right ) }{21\,d\sin \left ( dx+c \right ) }\sqrt{\cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.80294, size = 436, normalized size = 2.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61995, size = 239, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (3 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} \cos \left (d x + c\right )^{2} + 23 \, a^{2} \cos \left (d x + c\right ) + 46 \, a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{21 \,{\left (d \cos \left (d x + c\right ) + d\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 \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \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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