Integrand size = 35, antiderivative size = 322 \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\frac {32 a^3 (4184 A+4615 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (4184 A+4615 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d} \]
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Time = 1.36 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3040, 3054, 3059, 2851, 2850} \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\frac {2 a^3 (280 A+299 B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{1287 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {4 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{15015 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {32 a^3 (4184 A+4615 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (16 A+13 B) \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{143 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {13}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{13 d} \]
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Rule 2850
Rule 2851
Rule 3040
Rule 3054
Rule 3059
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {15}{2}}(c+d x)} \, dx \\ & = \frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {1}{13} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (16 A+13 B)+\frac {1}{2} a (8 A+13 B) \cos (c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {1}{143} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {1}{4} a^2 (280 A+299 B)+\frac {1}{4} a^2 (216 A+247 B) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (a^2 (4184 A+4615 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{1287} \\ & = \frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (2 a^2 (4184 A+4615 B) \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 {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (8 a^2 (4184 A+4615 B) \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 {16 a^3 (4184 A+4615 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (16 a^2 (4184 A+4615 B) \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 {32 a^3 (4184 A+4615 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (4184 A+4615 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.53 \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (171806 A+162955 B+35 (5552 A+5083 B) \cos (c+d x)+14 (15167 A+15925 B) \cos (2 (c+d x))+62760 A \cos (3 (c+d x))+69225 B \cos (3 (c+d x))+62760 A \cos (4 (c+d x))+69225 B \cos (4 (c+d x))+8368 A \cos (5 (c+d x))+9230 B \cos (5 (c+d x))+8368 A \cos (6 (c+d x))+9230 B \cos (6 (c+d x))) \sec ^{\frac {13}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{90090 d} \]
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Time = 2.20 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.51
\[-\frac {2 a^{2} \cot \left (d x +c \right ) \left (\cos \left (d x +c \right )-1\right ) \left (\left (66944 \left (\cos ^{6}\left (d x +c \right )\right )+33472 \left (\cos ^{5}\left (d x +c \right )\right )+25104 \left (\cos ^{4}\left (d x +c \right )\right )+20920 \left (\cos ^{3}\left (d x +c \right )\right )+18305 \left (\cos ^{2}\left (d x +c \right )\right )+11970 \cos \left (d x +c \right )+3465\right ) A +\cos \left (d x +c \right ) \left (73840 \left (\cos ^{5}\left (d x +c \right )\right )+36920 \left (\cos ^{4}\left (d x +c \right )\right )+27690 \left (\cos ^{3}\left (d x +c \right )\right )+23075 \left (\cos ^{2}\left (d x +c \right )\right )+14560 \cos \left (d x +c \right )+4095\right ) B \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\sec ^{\frac {15}{2}}\left (d x +c \right )\right )}{45045 d}\]
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Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.55 \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\frac {2 \, {\left (16 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} + 8 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 6 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (523 \, A + 416 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 315 \, {\left (38 \, A + 13 \, B\right )} a^{2} \cos \left (d x + c\right ) + 3465 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )} \sqrt {\cos \left (d x + c\right )}} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (280) = 560\).
Time = 0.37 (sec) , antiderivative size = 763, normalized size of antiderivative = 2.37 \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\text {Timed out} \]
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Time = 6.74 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.45 \[ \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{2}}(c+d x) \, dx=\text {Too large to display} \]
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