Integrand size = 33, antiderivative size = 211 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (33 A+25 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a (33 A+25 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac {2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d} \]
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Time = 0.48 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3125, 3047, 3102, 2830, 2726, 2725} \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (33 A+25 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{693 d}+\frac {2 (99 A+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac {2 a (33 A+25 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac {10 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d} \]
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Rule 2725
Rule 2726
Rule 2830
Rule 3047
Rule 3102
Rule 3125
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {2 \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (11 A+4 C)+\frac {5}{2} a C \cos (c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {2 \int (a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (11 A+4 C) \cos (c+d x)+\frac {5}{2} a C \cos ^2(c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac {4 \int (a+a \cos (c+d x))^{5/2} \left (\frac {35 a^2 C}{4}+\frac {1}{4} a^2 (99 A+26 C) \cos (c+d x)\right ) \, dx}{99 a^2} \\ & = \frac {2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac {1}{231} (5 (33 A+25 C)) \int (a+a \cos (c+d x))^{5/2} \, dx \\ & = \frac {2 a (33 A+25 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac {2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac {1}{231} (8 a (33 A+25 C)) \int (a+a \cos (c+d x))^{3/2} \, dx \\ & = \frac {16 a^2 (33 A+25 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a (33 A+25 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac {2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac {1}{693} \left (32 a^2 (33 A+25 C)\right ) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (33 A+25 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a (33 A+25 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{231 d}+\frac {2 (99 A+26 C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{693 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac {10 C (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.55 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (27456 A+22928 C+2 (6666 A+6989 C) \cos (c+d x)+16 (198 A+325 C) \cos (2 (c+d x))+396 A \cos (3 (c+d x))+1735 C \cos (3 (c+d x))+448 C \cos (4 (c+d x))+63 C \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{5544 d} \]
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Time = 12.99 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.65
method | result | size |
default | \(\frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-504 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2156 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-198 A -3762 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (693 A +3465 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-924 A -1848 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+693 A +693 C \right ) \sqrt {2}}{693 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(137\) |
parts | \(\frac {8 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (6 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8\right ) \sqrt {2}}{21 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {8 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (504 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-364 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+178 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+100 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200\right ) \sqrt {2}}{693 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(200\) |
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Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.61 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (63 \, C a^{2} \cos \left (d x + c\right )^{5} + 224 \, C a^{2} \cos \left (d x + c\right )^{4} + {\left (99 \, A + 355 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 6 \, {\left (66 \, A + 71 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (759 \, A + 568 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (759 \, A + 568 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{693 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.44 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {132 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + {\left (63 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1287 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 3465 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 8778 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 31878 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{11088 \, d} \]
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Time = 2.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.13 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (63 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 99 \, {\left (4 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 693 \, {\left (4 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 462 \, {\left (22 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 19 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 1386 \, {\left (30 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 23 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{11088 \, d} \]
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Timed out. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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