Optimal. Leaf size=334 \[ \frac {16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+9230 B+10439 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+9230 B+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2522 B+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+182 B+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (5 A+13 B) \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d} \]
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Rubi [A]
time = 0.76, 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 = {4350, 4171,
4102, 4100, 3890, 3889} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3889
Rule 3890
Rule 4100
Rule 4102
Rule 4171
Rule 4350
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 = 6.74, size = 244, normalized size = 0.73 \begin {gather*} \frac {\cos ^{\frac {9}{2}}(c+d x) \sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{16} (21 A+23 B+26 C) \sin \left (\frac {1}{2} (c+d x)\right )+\frac {1}{192} (71 A+76 B+80 C) \sin \left (\frac {3}{2} (c+d x)\right )+\frac {1}{320} (51 A+50 B+48 C) \sin \left (\frac {5}{2} (c+d x)\right )+\frac {1}{224} (15 A+13 B+10 C) \sin \left (\frac {7}{2} (c+d x)\right )+\frac {1}{288} (7 A+5 B+2 C) \sin \left (\frac {9}{2} (c+d x)\right )+\frac {1}{704} (5 A+2 B) \sin \left (\frac {11}{2} (c+d x)\right )+\frac {1}{832} A \sin \left (\frac {13}{2} (c+d x)\right )\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 222, normalized size = 0.66
method | result | size |
default | \(-\frac {2 a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (3465 A \left (\cos ^{6}\left (d x +c \right )\right )+11970 A \left (\cos ^{5}\left (d x +c \right )\right )+4095 B \left (\cos ^{5}\left (d x +c \right )\right )+18305 A \left (\cos ^{4}\left (d x +c \right )\right )+14560 B \left (\cos ^{4}\left (d x +c \right )\right )+5005 C \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+18590 C \left (\cos ^{3}\left (d x +c \right )\right )+25104 A \left (\cos ^{2}\left (d x +c \right )\right )+27690 B \left (\cos ^{2}\left (d x +c \right )\right )+31317 C \left (\cos ^{2}\left (d x +c \right )\right )+33472 A \cos \left (d x +c \right )+36920 B \cos \left (d x +c \right )+41756 C \cos \left (d x +c \right )+66944 A +73840 B +83512 C \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{45045 d \sin \left (d x +c \right )}\) | \(222\) |
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. 1135 vs.
\(2 (292) = 584\).
time = 0.72, size = 1135, normalized size = 3.40 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.39, size = 189, normalized size = 0.57 \begin {gather*} \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 )}} \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 )}^{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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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