Optimal. Leaf size=219 \[ \frac{2 a^2 (52 A+63 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{21 d} \]
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Rubi [A] time = 0.722911, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {4265, 4087, 4017, 4015, 3805, 3804} \[ \frac{2 a^2 (52 A+63 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{21 d} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4087
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} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{3 a A}{2}+\frac{1}{2} a (4 A+9 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{4} a^2 (52 A+63 C)+\frac{1}{4} a^2 (40 A+63 C) \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac{2 a^2 (52 A+63 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{1}{105} \left (a (136 A+189 C) \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{2 a^2 (136 A+189 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (52 A+63 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{1}{315} \left (2 a (136 A+189 C) \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{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (136 A+189 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (52 A+63 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 1.30907, size = 103, normalized size = 0.47 \[ \frac{a \sqrt{\cos (c+d x)} \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} (2 (799 A+756 C) \cos (c+d x)+4 (137 A+63 C) \cos (2 (c+d x))+170 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+2689 A+3276 C)}{1260 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.332, size = 120, 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}+102\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+63\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+136\,A\cos \left ( dx+c \right ) +189\,C\cos \left ( dx+c \right ) +272\,A+378\,C \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 = 2.19762, size = 734, 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 = 0.499324, size = 321, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (35 \, A a \cos \left (d x + c\right )^{4} + 85 \, A a \cos \left (d x + c\right )^{3} + 3 \,{\left (34 \, A + 21 \, C\right )} a \cos \left (d x + c\right )^{2} +{\left (136 \, A + 189 \, C\right )} a \cos \left (d x + c\right ) + 2 \,{\left (136 \, A + 189 \, C\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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