Optimal. Leaf size=273 \[ \frac {2 a^3 (2717 A+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (10439 A+8368 C) \sin (c+d x)}{6435 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (143 A+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{1287 d}-\frac {4 a^2 (10439 A+8368 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{45045 d}+\frac {2 a (10439 A+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac {10 a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]
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Rubi [A] time = 0.86, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3046, 2976, 2981, 2759, 2751, 2646} \[ \frac {2 a^3 (2717 A+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (143 A+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{1287 d}+\frac {2 a^3 (10439 A+8368 C) \sin (c+d x)}{6435 d \sqrt {a \cos (c+d x)+a}}-\frac {4 a^2 (10439 A+8368 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{45045 d}+\frac {2 a (10439 A+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac {10 a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rule 2976
Rule 2981
Rule 3046
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {2 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (13 A+6 C)+\frac {5}{2} a C \cos (c+d x)\right ) \, dx}{13 a}\\ &=\frac {10 a C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {4 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {1}{4} a^2 (143 A+96 C)+\frac {1}{4} a^2 (143 A+136 C) \cos (c+d x)\right ) \, dx}{143 a}\\ &=\frac {2 a^2 (143 A+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {8 \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {15}{8} a^3 (143 A+112 C)+\frac {1}{8} a^3 (2717 A+2224 C) \cos (c+d x)\right ) \, dx}{1287 a}\\ &=\frac {2 a^3 (2717 A+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (143 A+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (a^2 (10439 A+8368 C)\right ) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (2717 A+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (143 A+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (10439 A+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac {10 a C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {(2 a (10439 A+8368 C)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{15015}\\ &=\frac {2 a^3 (2717 A+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a^2 (10439 A+8368 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{45045 d}+\frac {2 a^2 (143 A+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (10439 A+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac {10 a C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (a^2 (10439 A+8368 C)\right ) \int \sqrt {a+a \cos (c+d x)} \, dx}{6435}\\ &=\frac {2 a^3 (10439 A+8368 C) \sin (c+d x)}{6435 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2717 A+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a^2 (10439 A+8368 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{45045 d}+\frac {2 a^2 (143 A+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (10439 A+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac {10 a C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}\\ \end {align*}
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Mathematica [A] time = 1.29, size = 138, normalized size = 0.51 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (8 (222794 A+226573 C) \cos (c+d x)+(581152 A+746519 C) \cos (2 (c+d x))+148720 A \cos (3 (c+d x))+20020 A \cos (4 (c+d x))+3233516 A+287060 C \cos (3 (c+d x))+94010 C \cos (4 (c+d x))+23940 C \cos (5 (c+d x))+3465 C \cos (6 (c+d x))+2798182 C)}{720720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 151, normalized size = 0.55 \[ \frac {2 \, {\left (3465 \, C a^{2} \cos \left (d x + c\right )^{6} + 11970 \, C a^{2} \cos \left (d x + c\right )^{5} + 35 \, {\left (143 \, A + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (1859 \, A + 2092 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (10439 \, A + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 4 \, {\left (10439 \, A + 8368 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (10439 \, A + 8368 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.97, size = 345, normalized size = 1.26 \[ \frac {1}{1441440} \, \sqrt {2} {\left (\frac {3465 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {13}{2} \, d x + \frac {13}{2} \, c\right )}{d} + \frac {20475 \, 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 )}{d} + \frac {10010 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {64350 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, 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 )}{d} + \frac {27027 \, {\left (16 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 17 \, 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 )}{d} + \frac {15015 \, {\left (80 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 71 \, 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 )}{d} + \frac {180180 \, {\left (12 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, 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 )}{d} + \frac {2522520 \, {\left (A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 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 )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 156, normalized size = 0.57 \[ \frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (55440 C \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-262080 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20020 A +520520 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-77220 A -566280 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (117117 A +369369 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-90090 A -150150 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45045 A +45045 C \right ) \sqrt {2}}{45045 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 223, normalized size = 0.82 \[ \frac {572 \, {\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + {\left (3465 \, \sqrt {2} a^{2} \sin \left (\frac {13}{2} \, d x + \frac {13}{2} \, c\right ) + 20475 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 70070 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 193050 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 459459 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1066065 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3783780 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{1441440 \, d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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