Optimal. Leaf size=223 \[ \frac {2 a (99 A+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {4 (99 A+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac {8 (99 A+80 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}+\frac {2 a C \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.51, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3046, 2981, 2770, 2759, 2751, 2646} \[ \frac {2 a (99 A+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {4 (99 A+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac {8 (99 A+80 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {2 C \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d}+\frac {2 a C \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rule 2770
Rule 2981
Rule 3046
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (A+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 C \cos (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 a 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+80 C) \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a (99 A+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a 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+80 C)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a (99 A+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a 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+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}+\frac {(4 (99 A+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+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a C \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 (99 A+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+80 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 a d}+\frac {1}{495} (2 (99 A+80 C)) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (99 A+80 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a C \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 (99 A+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+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 = 0.97, size = 114, normalized size = 0.51 \[ \frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (2 (9306 A+9095 C) \cos (c+d x)+16 (297 A+415 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+30096 A+3175 C \cos (3 (c+d x))+700 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+26420 C)}{27720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 110, normalized size = 0.49 \[ \frac {2 \, {\left (315 \, C \cos \left (d x + c\right )^{5} + 350 \, C \cos \left (d x + c\right )^{4} + 5 \, {\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right ) + 1584 \, A + 1280 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 223, normalized size = 1.00 \[ \frac {1}{55440} \, \sqrt {2} {\left (\frac {315 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {385 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {495 \, {\left (4 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {693 \, {\left (4 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {2310 \, {\left (6 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {6930 \, {\left (6 \, A \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 135, normalized size = 0.61 \[ \frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-10080 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+30800 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3960 A -39600 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8316 A +27720 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6930 A -11550 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 C \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 160, normalized size = 0.72 \[ \frac {396 \, {\left (5 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 7 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 35 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 105 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 5 \, {\left (63 \, \sqrt {2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 77 \, \sqrt {2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 495 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 693 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2310 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 6930 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{55440 \, d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )} \,d x \]
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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