Optimal. Leaf size=220 \[ \frac{2 a (8 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (8 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a (8 A+9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{32 a (8 A+9 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.475591, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2955, 4015, 3805, 3804} \[ \frac{2 a (8 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (8 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a (8 A+9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{32 a (8 A+9 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2955
Rule 4015
Rule 3805
Rule 3804
Rubi steps
\begin{align*} \int \cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{9} \left ((8 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a (8 A+9 B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{21} \left (2 (8 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{4 a (8 A+9 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (8 A+9 B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{105} \left (8 (8 A+9 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{16 a (8 A+9 B) \sqrt{\cos (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{4 a (8 A+9 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (8 A+9 B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{315} \left (16 (8 A+9 B) \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{32 a (8 A+9 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{16 a (8 A+9 B) \sqrt{\cos (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{4 a (8 A+9 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (8 A+9 B) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.484972, size = 119, normalized size = 0.54 \[ \frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} (94 (8 A+9 B) \cos (c+d x)+4 (83 A+54 B) \cos (2 (c+d x))+80 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+1321 A+90 B \cos (3 (c+d x))+1368 B)}{1260 d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.34, size = 130, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 35\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+40\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+45\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+48\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+54\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+64\,A\cos \left ( dx+c \right ) +72\,B\cos \left ( dx+c \right ) +128\,A+144\,B \right ) }{315\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}\sqrt{\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.06703, size = 738, normalized size = 3.35 \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 = 0.485765, size = 309, normalized size = 1.4 \begin{align*} \frac{2 \,{\left (35 \, A \cos \left (d x + c\right )^{4} + 5 \,{\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \,{\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right ) + 128 \, A + 144 \, B\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 )}{315 \,{\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(-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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