Optimal. Leaf size=228 \[ \frac{2 a^2 (10 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (34 A+39 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d} \]
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Rubi [A] time = 0.686358, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2955, 4017, 4015, 3805, 3804} \[ \frac{2 a^2 (10 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (34 A+39 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d} \]
Antiderivative was successfully verified.
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Rule 2955
Rule 4017
Rule 4015
Rule 3805
Rule 3804
Rubi steps
\begin{align*} \int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} (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) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d}+\frac{1}{9} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{2} a (10 A+9 B)+\frac{3}{2} a (2 A+3 B) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (10 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) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d}+\frac{1}{21} \left (a (34 A+39 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{2 a^2 (34 A+39 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (10 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) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d}+\frac{1}{105} \left (4 a (34 A+39 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{8 a^2 (34 A+39 B) \sqrt{\cos (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (34 A+39 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (10 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) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d}+\frac{1}{315} \left (8 a (34 A+39 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{16 a^2 (34 A+39 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{8 a^2 (34 A+39 B) \sqrt{\cos (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (34 A+39 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (10 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) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.422066, size = 118, normalized size = 0.52 \[ \frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left (5 (17 A+9 B) \cos ^3(c+d x)+3 (34 A+39 B) \cos ^2(c+d x)+4 (34 A+39 B) \cos (c+d x)+8 (34 A+39 B)+35 A \cos ^4(c+d x)\right )}{315 d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.315, size = 131, normalized size = 0.6 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 35\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+85\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+45\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+102\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+117\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+136\,A\cos \left ( dx+c \right ) +156\,B\cos \left ( dx+c \right ) +272\,A+312\,B \right ) }{315\,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 = 2.09384, size = 753, normalized size = 3.3 \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 = 0.48523, size = 332, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (35 \, A a \cos \left (d x + c\right )^{4} + 5 \,{\left (17 \, A + 9 \, B\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (34 \, A + 39 \, B\right )} a \cos \left (d x + c\right )^{2} + 4 \,{\left (34 \, A + 39 \, B\right )} a \cos \left (d x + c\right ) + 8 \,{\left (34 \, A + 39 \, B\right )} a\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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