Optimal. Leaf size=334 \[ \frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{9009 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15015 d \sqrt {a \sec (c+d x)+a}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{45045 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (136 A+182 B+143 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{1287 d}+\frac {2 a (5 A+13 B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {11}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d} \]
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Rubi [A] time = 1.10, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {2 a^2 (136 A+182 B+143 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{1287 d}+\frac {2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{9009 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15015 d \sqrt {a \sec (c+d x)+a}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{45045 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a (5 A+13 B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {11}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 3804
Rule 3805
Rule 4015
Rule 4017
Rule 4086
Rule 4265
Rubi steps
\begin {align*} \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \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))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx\\ &=\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 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 {1}{2} a (5 A+13 B)+\frac {1}{2} a (6 A+13 C) \sec (c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx}{13 a}\\ &=\frac {2 a (5 A+13 B) \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 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 {1}{4} a^2 (136 A+182 B+143 C)+\frac {1}{4} a^2 (96 A+78 B+143 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{143 a}\\ &=\frac {2 a^2 (136 A+182 B+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (5 A+13 B) \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 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 {1}{8} a^3 (2224 A+2522 B+2717 C)+\frac {3}{8} a^3 (560 A+598 B+715 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{1287 a}\\ &=\frac {2 a^3 (2224 A+2522 B+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (5 A+13 B) \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (a^2 (8368 A+9230 B+10439 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}{3003}\\ &=\frac {2 a^3 (8368 A+9230 B+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2522 B+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (5 A+13 B) \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (4 a^2 (8368 A+9230 B+10439 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}{15015}\\ &=\frac {8 a^3 (8368 A+9230 B+10439 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2522 B+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (5 A+13 B) \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (8 a^2 (8368 A+9230 B+10439 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}{45045}\\ &=\frac {16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2522 B+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (5 A+13 B) \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}\\ \end {align*}
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Mathematica [A] time = 2.49, size = 190, normalized size = 0.57 \[ \frac {a^2 \sqrt {\cos (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sec (c+d x)+1)} (4 (453146 A+454285 B+445588 C) \cos (c+d x)+(746519 A+676000 B+581152 C) \cos (2 (c+d x))+287060 A \cos (3 (c+d x))+94010 A \cos (4 (c+d x))+23940 A \cos (5 (c+d x))+3465 A \cos (6 (c+d x))+2798182 A+225550 B \cos (3 (c+d x))+58240 B \cos (4 (c+d x))+8190 B \cos (5 (c+d x))+2980640 B+148720 C \cos (3 (c+d x))+20020 C \cos (4 (c+d x))+3233516 C)}{720720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 189, normalized size = 0.57 \[ \frac {2 \, {\left (3465 \, A a^{2} \cos \left (d x + c\right )^{6} + 315 \, {\left (38 \, A + 13 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 35 \, {\left (523 \, A + 416 \, B + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (4184 \, A + 4615 \, B + 3718 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 4 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (8368 \, A + 9230 \, B + 10439 \, C\right )} a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {13}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.04, size = 222, normalized size = 0.66 \[ -\frac {2 a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (3465 A \left (\cos ^{6}\left (d x +c \right )\right )+11970 A \left (\cos ^{5}\left (d x +c \right )\right )+4095 B \left (\cos ^{5}\left (d x +c \right )\right )+18305 A \left (\cos ^{4}\left (d x +c \right )\right )+14560 B \left (\cos ^{4}\left (d x +c \right )\right )+5005 C \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+18590 C \left (\cos ^{3}\left (d x +c \right )\right )+25104 A \left (\cos ^{2}\left (d x +c \right )\right )+27690 B \left (\cos ^{2}\left (d x +c \right )\right )+31317 C \left (\cos ^{2}\left (d x +c \right )\right )+33472 A \cos \left (d x +c \right )+36920 B \cos \left (d x +c \right )+41756 C \cos \left (d x +c \right )+66944 A +73840 B +83512 C \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{45045 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.86, size = 1135, normalized size = 3.40 \[ \text {result too large to display} \]
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 \[ \int {\cos \left (c+d\,x\right )}^{13/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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