Optimal. Leaf size=266 \[ \frac {16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{1155 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (112 A+143 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{385 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (28 A+33 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{33 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d} \]
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Rubi [A]
time = 0.55, antiderivative size = 266, normalized size of antiderivative = 1.00, 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 = {4350, 4172,
4102, 4100, 3890, 3889} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3889
Rule 3890
Rule 4100
Rule 4102
Rule 4172
Rule 4350
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*}
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Mathematica [A]
time = 2.03, size = 125, normalized size = 0.47 \begin {gather*} \frac {a \sqrt {\cos (c+d x)} (18494 A+21736 C+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))+660 C \cos (3 (c+d x))+490 A \cos (4 (c+d x))+105 A \cos (5 (c+d x))) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{9240 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 142, normalized size = 0.53
method | result | size |
default | \(-\frac {2 a \left (-1+\cos \left (d x +c \right )\right ) \left (105 A \left (\cos ^{5}\left (d x +c \right )\right )+245 A \left (\cos ^{4}\left (d x +c \right )\right )+280 A \left (\cos ^{3}\left (d x +c \right )\right )+165 C \left (\cos ^{3}\left (d x +c \right )\right )+336 A \left (\cos ^{2}\left (d x +c \right )\right )+429 C \left (\cos ^{2}\left (d x +c \right )\right )+448 A \cos \left (d x +c \right )+572 C \cos \left (d x +c \right )+896 A +1144 C \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{1155 d \sin \left (d x +c \right )}\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 652 vs.
\(2 (230) = 460\).
time = 0.67, size = 652, normalized size = 2.45 \begin {gather*} \frac {7 \, \sqrt {2} {\left (3630 \, a \cos \left (\frac {10}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 990 \, a \cos \left (\frac {8}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 429 \, a \cos \left (\frac {6}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 165 \, a \cos \left (\frac {4}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, a \cos \left (\frac {2}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) - 3630 \, a \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \sin \left (\frac {10}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) - 990 \, a \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \sin \left (\frac {8}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) - 429 \, a \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \sin \left (\frac {6}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) - 165 \, a \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \sin \left (\frac {4}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) - 55 \, a \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \sin \left (\frac {2}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 30 \, a \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, a \sin \left (\frac {9}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 165 \, a \sin \left (\frac {7}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 429 \, a \sin \left (\frac {5}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 990 \, a \sin \left (\frac {3}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right ) + 3630 \, a \sin \left (\frac {1}{11} \, \arctan \left (\sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ), \cos \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )\right )\right )\right )} A \sqrt {a} - 44 \, \sqrt {2} {\left (175 \, a \cos \left (\frac {7}{4} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right )\right )\right ) \sin \left (2 \, d x + 2 \, c\right ) - 5 \, {\left (35 \, a \cos \left (2 \, d x + 2 \, c\right ) + 6 \, a\right )} \sin \left (\frac {7}{4} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right )\right )\right ) - 126 \, a \sin \left (\frac {5}{4} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right )\right )\right ) - 175 \, a \sin \left (\frac {3}{4} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right )\right )\right ) - 1470 \, a \sin \left (\frac {1}{4} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right )\right )\right )\right )} C \sqrt {a}}{36960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.34, size = 136, normalized size = 0.51 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^{11/2}\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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