Optimal. Leaf size=266 \[ \frac{2 a^2 (28 A+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{231 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{385 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (112 A+143 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{33 d} \]
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Rubi [A] time = 0.806646, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 a^2 (28 A+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{231 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{385 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (112 A+143 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{33 d} \]
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{11}{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{11}{2}}(c+d x)} \, dx\\ &=\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 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 (6 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{33 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 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{3}{4} a^2 (28 A+33 C)+\frac{9}{4} a^2 (8 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 a^2 (28 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{231 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)}{33 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{1}{77} \left (a (112 A+143 C) \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 (112 A+143 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (28 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{231 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)}{33 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{1}{385} \left (4 a (112 A+143 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{8 a^2 (112 A+143 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{1155 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (112 A+143 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (28 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{231 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)}{33 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac{\left (8 a (112 A+143 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}{1155}\\ &=\frac{16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{8 a^2 (112 A+143 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{1155 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (112 A+143 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (28 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{231 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)}{33 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 1.99648, size = 125, normalized size = 0.47 \[ \frac{a \sqrt{\cos (c+d x)} \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} (2 (5789 A+5566 C) \cos (c+d x)+8 (581 A+429 C) \cos (2 (c+d x))+1645 A \cos (3 (c+d x))+490 A \cos (4 (c+d x))+105 A \cos (5 (c+d x))+18494 A+660 C \cos (3 (c+d x))+21736 C)}{9240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.351, size = 142, normalized size = 0.5 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 105\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+245\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+280\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+165\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+336\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+429\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+448\,A\cos \left ( dx+c \right ) +572\,C\cos \left ( dx+c \right ) +896\,A+1144\,C \right ) }{1155\,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 = 2.24813, size = 880, normalized size = 3.31 \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 = 0.499507, size = 378, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (105 \, A a \cos \left (d x + c\right )^{5} + 245 \, A a \cos \left (d x + c\right )^{4} + 5 \,{\left (56 \, A + 33 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \,{\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right ) + 8 \,{\left (112 \, A + 143 \, 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 )}{1155 \,{\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 (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{\frac{11}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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