Optimal. Leaf size=234 \[ \frac {4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 a (187 A+168 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d} \]
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Rubi [A]
time = 0.33, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3055, 3060,
2849, 2838, 2830, 2725} \begin {gather*} \frac {2 a^2 (11 A+12 B) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (187 A+168 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {4 (187 A+168 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac {8 a (187 A+168 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {2 a B \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2830
Rule 2838
Rule 2849
Rule 3055
Rule 3060
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {1}{2} a (11 A+8 B)+\frac {1}{2} a (11 A+12 B) \cos (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{99} (a (187 A+168 B)) \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{231} (2 a (187 A+168 B)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {(4 (187 A+168 B)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{1155}\\ &=\frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 a (187 A+168 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {1}{495} (2 a (187 A+168 B)) \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 a (187 A+168 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 125, normalized size = 0.53 \begin {gather*} \frac {a \sqrt {a (1+\cos (c+d x))} (59158 A+55482 B+(35156 A+34734 B) \cos (c+d x)+8 (1507 A+1743 B) \cos (2 (c+d x))+3740 A \cos (3 (c+d x))+4935 B \cos (3 (c+d x))+770 A \cos (4 (c+d x))+1470 B \cos (4 (c+d x))+315 B \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{27720 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 142, normalized size = 0.61
method | result | size |
default | \(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-5040 B \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3080 A +18480 B \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-9900 A -27720 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12474 A +22176 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-8085 A -10395 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 B \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 185, normalized size = 0.79 \begin {gather*} \frac {22 \, {\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 21 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 165 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 429 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 990 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3630 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a}}{55440 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 125, normalized size = 0.53 \begin {gather*} \frac {2 \, {\left (315 \, B a \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, A + 21 \, B\right )} a \cos \left (d x + c\right )^{4} + 5 \, {\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right )^{3} + 6 \, {\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right )^{2} + 8 \, {\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right ) + 16 \, {\left (187 \, A + 168 \, B\right )} a\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.23, size = 235, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} {\left (315 \, B a \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 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, B a \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 ) + 495 \, {\left (6 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, B a \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 (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, B a \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 ) + 2310 \, {\left (10 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 9 \, B a \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 ) + 6930 \, {\left (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 11 \, B a \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}}{55440 \, d} \end {gather*}
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 \begin {gather*} \int {\cos \left (c+d\,x\right )}^3\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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