Optimal. Leaf size=284 \[ \frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d} \]
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Rubi [A] time = 0.885488, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4265, 4086, 4017, 4015, 3805, 3804} \[ \frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d} \]
Antiderivative was successfully verified.
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Rule 4265
Rule 4086
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+B \sec (c+d x)+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+B \sec (c+d x)+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{1}{2} a (3 A+11 B)+\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 (3 A+11 B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 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{1}{4} a^2 (84 A+110 B+99 C)+\frac{3}{4} a^2 (24 A+22 B+33 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 a^2 (84 A+110 B+99 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 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}{231} \left (a (336 A+374 B+429 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 (336 A+374 B+429 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (84 A+110 B+99 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 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 a (336 A+374 B+429 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}{1155}\\ &=\frac{8 a^2 (336 A+374 B+429 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (336 A+374 B+429 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (84 A+110 B+99 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 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 (336 A+374 B+429 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}{3465}\\ &=\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{8 a^2 (336 A+374 B+429 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (336 A+374 B+429 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (84 A+110 B+99 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 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 = 2.10568, size = 158, normalized size = 0.56 \[ \frac{a \sqrt{\cos (c+d x)} \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} ((34734 A+44 (799 B+759 C)) \cos (c+d x)+8 (1743 A+1507 B+1287 C) \cos (2 (c+d x))+4935 A \cos (3 (c+d x))+1470 A \cos (4 (c+d x))+315 A \cos (5 (c+d x))+55482 A+3740 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+59158 B+1980 C \cos (3 (c+d x))+65208 C)}{27720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.284, size = 187, normalized size = 0.7 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 315\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+735\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+385\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+840\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+935\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+495\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1008\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1122\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1287\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1344\,A\cos \left ( dx+c \right ) +1496\,B\cos \left ( dx+c \right ) +1716\,C\cos \left ( dx+c \right ) +2688\,A+2992\,B+3432\,C \right ) }{3465\,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 = 2.49258, size = 1164, normalized size = 4.1 \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 = 0.511192, size = 437, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (315 \, A a \cos \left (d x + c\right )^{5} + 35 \,{\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right )^{4} + 5 \,{\left (168 \, A + 187 \, B + 99 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right ) + 8 \,{\left (336 \, A + 374 \, B + 429 \, 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 )}{3465 \,{\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} + B \sec \left (d x + c\right ) + 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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