Optimal. Leaf size=229 \[ \frac {64 a^3 (165 A+143 B+125 C) \sin (c+d x)}{3465 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (165 A+143 B+125 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {2 (99 A-22 B+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac {2 a (165 A+143 B+125 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac {2 (11 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d} \]
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Rubi [A] time = 0.49, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3045, 2968, 3023, 2751, 2647, 2646} \[ \frac {16 a^2 (165 A+143 B+125 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {64 a^3 (165 A+143 B+125 C) \sin (c+d x)}{3465 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (99 A-22 B+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac {2 a (165 A+143 B+125 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac {2 (11 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rule 2968
Rule 3023
Rule 3045
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\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 {1}{2} a (11 B+5 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 {1}{2} a (11 B+5 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 {2 (11 B+5 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 {7}{4} a^2 (11 B+5 C)+\frac {1}{4} a^2 (99 A-22 B+26 C) \cos (c+d x)\right ) \, dx}{99 a^2}\\ &=\frac {2 (99 A-22 B+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 {2 (11 B+5 C) (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac {1}{231} (165 A+143 B+125 C) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac {2 a (165 A+143 B+125 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 (99 A-22 B+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 {2 (11 B+5 C) (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac {(8 a (165 A+143 B+125 C)) \int (a+a \cos (c+d x))^{3/2} \, dx}{1155}\\ &=\frac {16 a^2 (165 A+143 B+125 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a (165 A+143 B+125 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 (99 A-22 B+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 {2 (11 B+5 C) (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}+\frac {\left (32 a^2 (165 A+143 B+125 C)\right ) \int \sqrt {a+a \cos (c+d x)} \, dx}{3465}\\ &=\frac {64 a^3 (165 A+143 B+125 C) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (165 A+143 B+125 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a (165 A+143 B+125 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 (99 A-22 B+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 {2 (11 B+5 C) (a+a \cos (c+d x))^{7/2} \sin (c+d x)}{99 a d}\\ \end {align*}
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Mathematica [A] time = 1.26, size = 147, normalized size = 0.64 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} ((66660 A+68552 B+69890 C) \cos (c+d x)+16 (990 A+1397 B+1625 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+137280 A+5720 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+124366 B+8675 C \cos (3 (c+d x))+2240 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+114640 C)}{27720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 148, normalized size = 0.65 \[ \frac {2 \, {\left (315 \, C a^{2} \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, B + 32 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (99 \, A + 286 \, B + 355 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (660 \, A + 803 \, B + 710 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (3795 \, A + 3212 \, B + 2840 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (3795 \, A + 3212 \, B + 2840 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 336, normalized size = 1.47 \[ \frac {1}{55440} \, \sqrt {2} {\left (\frac {315 \, 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 {385 \, {\left (2 \, B 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 {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {495 \, {\left (4 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 10 \, B 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 )}{d} + \frac {693 \, {\left (20 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 24 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 25 \, 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 {2310 \, {\left (22 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 20 \, B 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 )}{d} + \frac {6930 \, {\left (30 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 26 \, B 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 )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 154, normalized size = 0.67 \[ \frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-2520 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1540 B +10780 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-990 A -5940 B -18810 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3465 A +9009 B +17325 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-4620 A -6930 B -9240 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 B +3465 C \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 282, normalized size = 1.23 \[ \frac {660 \, {\left (3 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 21 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 77 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 315 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 22 \, {\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + 5 \, {\left (63 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1287 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 3465 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 8778 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 31878 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{55440 \, d} \]
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 \[ \int \cos \left (c+d\,x\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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