Optimal. Leaf size=266 \[ \frac{2 a^2 (32 A+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{231 d}+\frac{2 a^3 (232 A+297 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d}+\frac{10 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d} \]
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Rubi [A] time = 0.937103, 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} \[ \frac{2 a^2 (32 A+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{231 d}+\frac{2 a^3 (232 A+297 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d}+\frac{10 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d} \]
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))^{5/2} \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))^{5/2} \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))^{5/2} \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))^{5/2} \left (\frac{5 a A}{2}+\frac{1}{2} a (4 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{10 a A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{3}{4} a^2 (32 A+33 C)+\frac{1}{4} a^2 (56 A+99 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 a^2 (32 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{10 a A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{5}{8} a^3 (232 A+297 C)+\frac{1}{8} a^3 (776 A+1089 C) \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac{2 a^3 (232 A+297 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (32 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{10 a A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{1}{231} \left (a^2 (568 A+759 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^3 (568 A+759 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^3 (232 A+297 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (32 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{10 a A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{1}{693} \left (2 a^2 (568 A+759 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^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a^3 (568 A+759 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^3 (232 A+297 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (32 A+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{10 a A \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 2.42434, size = 127, normalized size = 0.48 \[ \frac{a^2 \sqrt{\cos (c+d x)} \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} (2 (6989 A+6666 C) \cos (c+d x)+16 (325 A+198 C) \cos (2 (c+d x))+1735 A \cos (3 (c+d x))+448 A \cos (4 (c+d x))+63 A \cos (5 (c+d x))+22928 A+396 C \cos (3 (c+d x))+27456 C)}{5544 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.345, size = 144, normalized size = 0.5 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 63\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+224\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+355\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+99\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+426\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+396\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+568\,A\cos \left ( dx+c \right ) +759\,C\cos \left ( dx+c \right ) +1136\,A+1518\,C \right ) }{693\,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 = 2.19197, size = 938, normalized size = 3.53 \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 = 0.517164, size = 385, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (63 \, A a^{2} \cos \left (d x + c\right )^{5} + 224 \, A a^{2} \cos \left (d x + c\right )^{4} +{\left (355 \, A + 99 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 6 \,{\left (71 \, A + 66 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (568 \, A + 759 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (568 \, A + 759 \, C\right )} 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 )}{693 \,{\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{5}{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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