Optimal. Leaf size=313 \[ \frac {2 a^3 (2224 A+2717 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{15015 d \sqrt {a \cos (c+d x)+a}}+\frac {8 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (136 A+143 C) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{1287 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}+\frac {10 a A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d} \]
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Rubi [A] time = 1.06, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {4221, 3044, 2975, 2980, 2772, 2771} \[ \frac {2 a^2 (136 A+143 C) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{1287 d}+\frac {2 a^3 (2224 A+2717 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{15015 d \sqrt {a \cos (c+d x)+a}}+\frac {8 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {2 A \sin (c+d x) \sec ^{\frac {13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}+\frac {10 a A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d} \]
Antiderivative was successfully verified.
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Rule 2771
Rule 2772
Rule 2975
Rule 2980
Rule 3044
Rule 4221
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {15}{2}}(c+d x) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{5/2} \left (\frac {5 a A}{2}+\frac {1}{2} a (6 A+13 C) \cos (c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx}{13 a}\\ &=\frac {10 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {1}{4} a^2 (136 A+143 C)+\frac {1}{4} a^2 (96 A+143 C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{143 a}\\ &=\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {13}{2}}(c+d x) \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 \cos (c+d x)} \left (\frac {1}{8} a^3 (2224 A+2717 C)+\frac {15}{8} a^3 (112 A+143 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{1287 a}\\ &=\frac {2 a^3 (2224 A+2717 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (a^2 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (8368 A+10439 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2224 A+2717 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (4 a^2 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{15015}\\ &=\frac {8 a^3 (8368 A+10439 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2224 A+2717 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (8 a^2 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{45045}\\ &=\frac {16 a^3 (8368 A+10439 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {8 a^3 (8368 A+10439 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2224 A+2717 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 171, normalized size = 0.55 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \sqrt {a (\cos (c+d x)+1)} (1120 (347 A+286 C) \cos (c+d x)+14 (30334 A+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+343612 A+141570 C \cos (3 (c+d x))+156585 C \cos (4 (c+d x))+20878 C \cos (5 (c+d x))+20878 C \cos (6 (c+d x))+322751 C)}{180180 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 170, normalized size = 0.54 \[ \frac {2 \, {\left (8 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 4 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 3 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (2092 \, A + 1859 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (523 \, A + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 11970 \, A a^{2} \cos \left (d x + c\right ) + 3465 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )} \sqrt {\cos \left (d x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.62, size = 176, normalized size = 0.56 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (66944 A \left (\cos ^{6}\left (d x +c \right )\right )+83512 C \left (\cos ^{6}\left (d x +c \right )\right )+33472 A \left (\cos ^{5}\left (d x +c \right )\right )+41756 C \left (\cos ^{5}\left (d x +c \right )\right )+25104 A \left (\cos ^{4}\left (d x +c \right )\right )+31317 C \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+18590 C \left (\cos ^{3}\left (d x +c \right )\right )+18305 A \left (\cos ^{2}\left (d x +c \right )\right )+5005 C \left (\cos ^{2}\left (d x +c \right )\right )+11970 A \cos \left (d x +c \right )+3465 A \right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {15}{2}} a^{2}}{45045 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.93, size = 763, normalized size = 2.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.02, size = 897, normalized size = 2.87 \[ \text {result too large to display} \]
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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