Optimal. Leaf size=239 \[ \frac {2 a (99 A+88 B+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {4 (99 A+88 B+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac {8 (99 A+88 B+80 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {4 a (99 A+88 B+80 C) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d} \]
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Rubi [A] time = 0.55, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3045, 2981, 2770, 2759, 2751, 2646} \[ \frac {2 a (99 A+88 B+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {4 (99 A+88 B+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac {8 (99 A+88 B+80 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {4 a (99 A+88 B+80 C) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rule 2770
Rule 2981
Rule 3045
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {2 \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {1}{2} a (11 A+8 C)+\frac {1}{2} a (11 B+C) \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 a (11 B+C) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{99} (99 A+88 B+80 C) \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a (99 A+88 B+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+C) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{231} (2 (99 A+88 B+80 C)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a (99 A+88 B+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+C) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}+\frac {(4 (99 A+88 B+80 C)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{1155 a}\\ &=\frac {2 a (99 A+88 B+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+C) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 (99 A+88 B+80 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 C \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}+\frac {1}{495} (2 (99 A+88 B+80 C)) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {4 a (99 A+88 B+80 C) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (99 A+88 B+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+C) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 (99 A+88 B+80 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 C \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (99 A+88 B+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}\\ \end {align*}
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Mathematica [A] time = 1.30, size = 145, normalized size = 0.61 \[ \frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (2 (9306 A+8272 B+9095 C) \cos (c+d x)+8 (594 A+913 B+830 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+30096 A+1760 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+29062 B+3175 C \cos (3 (c+d x))+700 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+26420 C)}{27720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 128, normalized size = 0.54 \[ \frac {2 \, {\left (315 \, C \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (99 \, A + 88 \, B + 80 \, C\right )} \cos \left (d x + c\right ) + 1584 \, A + 1408 \, B + 1280 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.49, size = 290, normalized size = 1.21 \[ \frac {1}{55440} \, \sqrt {2} {\left (\frac {315 \, C \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 \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + C \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 \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \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 (4 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 8 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \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 (6 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 4 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \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 (6 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, B \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \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 = 0.67, size = 152, normalized size = 0.64 \[ \frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-10080 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6160 B +30800 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3960 A -15840 B -39600 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8316 A +16632 B +27720 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6930 A -9240 B -11550 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 B +3465 C \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 237, normalized size = 0.99 \[ \frac {396 \, {\left (5 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 7 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 35 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 105 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 22 \, {\left (35 \, \sqrt {2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 252 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 420 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 1890 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + 5 \, {\left (63 \, \sqrt {2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 77 \, \sqrt {2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 495 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 693 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2310 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 6930 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{55440 \, d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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